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Quantum Teleportation andMulti-photon Entanglement
Dissertation zur Erlangung des Grades einesDoktors der Naturwissenschaften
eingereicht von
M.Sc. Jian-Wei PanUniversity of Science and Technology of China
Durchgefuhrt am Institut fur Experimentalphysikder Universitat Wien
beio.Univ.Prof.Dr. Anton Zeilinger
Gefordert vom Fonds zur Forderung der wissenschaftlichen Forschung, Projekte
S6502 und F1506 und durch das TMR-Netzwerk The Physics of Quantum
Information der Europaischen Kommission.
Contents
1 Introduction 5
2 Manipulation of Entangled States 10
2.1 Quantum network and its applications . . . . . . . . . . . . . 11
2.2 Practical schemes for entangled-state analysis . . . . . . . . . 15
2.2.1 Bell-state analysis . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 GHZ-state analyzer . . . . . . . . . . . . . . . . . . . . 20
2.3 Polarization-entangled photon pairs . . . . . . . . . . . . . . . 29
3 Quantum Teleportation 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Quantum teleportation–the idea . . . . . . . . . . . . . . . . . 34
3.2.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 The concept of quantum teleportation . . . . . . . . . 35
3.3 Experimental teleportation . . . . . . . . . . . . . . . . . . . . 40
3.3.1 Experimental scheme . . . . . . . . . . . . . . . . . . . 40
3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Entanglement Swapping 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Theoretical scheme . . . . . . . . . . . . . . . . . . . . . . . . 54
i
ii CONTENTS
4.3 Experimental entanglement swapping . . . . . . . . . . . . . . 56
4.4 Generalization and applications . . . . . . . . . . . . . . . . . 61
5 Three-photon GHZ entanglement 64
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Observation of three-photon entanglement . . . . . . . . . . . 72
5.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . 75
6 Experimental tests of the GHZ theorem 76
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 The conflict with local realism . . . . . . . . . . . . . . . . . . 77
6.2.1 GHZ theorem . . . . . . . . . . . . . . . . . . . . . . . 77
6.2.2 Generalization to conditional GHZ state . . . . . . . . 81
6.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 84
6.4 Discussion and Prospects . . . . . . . . . . . . . . . . . . . . . 90
7 Conclusions and outlook 92
Zusammenfassung
Die vorliegende Dissertation ist das Ergebnis theoretischer und experimentellerArbeiten uber die Physik von Mehrteilcheninterferenz. Die theoretischenErgebnisse zeigen, daß man Quantenverschrankung mit einem Quantennet-zwerk aus einfachen Quantenlogikgattern und einer kleinen Anzahl von Qubitskontrollieren und manipulieren kann. Da es bis jetzt keine experimentelleDurchfuhrung von Quantengattern fur zwei unabhangig erzeugte Photonengibt, prasentieren wir hier eine realisierbare Methode, verschrankte Viel-teilchenzustande zu erzeugen und zu identifizieren.
In der experimentellen Arbeit wurden die zum Studium von neuarti-gen Vielteilcheninterferenzphanomenen notigen Techniken von Grund auf en-twickelt. Wir berichten in dieser Arbeit uber die erstmalige experimentelleRealisierung von Quantenteleportation, ’Entanglement Swapping’ und derErzeugung von Dreiteilchenverschrankung mithilfe einer gepulsten Quellefur polarisationsverschrankte Photonen. Mit der Quelle fur Dreiteilchen-verschrankung wurde das erste Experiment zum Test von lokalrealistischenTheorien ohne Ungleichungen durchgefuhrt.
Die in diesen Experimenten entwickelten Methoden sind von großer Be-deutung fur Forschungen auf dem Gebiet der Quanteninformation und furzukunftige fundamentale Experimente der Quantenmechanik.
1
Abstract
The present thesis is the result of theoretical and experimental work on thephysics of multiparticle interference. The theoretical results show that aquantum network with simple quantum logic gates and a handful of qubitsenables one to control and manipulate quantum entanglement. Because of thepresent absence of quantum gate for two independently produced photons,in the mean time we also present a practical way to generate and identifymultiparticle entangled state.
The experimental work has thoroughly developed the necessary tech-niques to study novel multiparticle interference phenomena. By making useof the pulsed source for polarization entangled photon pairs, in this thesis wereport for the first time the experimental realization of quantum teleporta-tion, of entanglement swapping and of production of three-particle entangle-ment. Using the three-particle entanglement source, here we also present thefirst experimental realization of a test of local realism without inequalities.
The methods developed in these experiments are of great significance bothfor exploring the field of quantum information and for future experiments onthe fundamental tests of quantum mechanics.
2
Acknowledgements
I am indebted to my advisor, Professor Anton Zeilinger, for his guidanceand support throughout the course of my work leading to this thesis. Hetaught me with wisdom, encouragement, rich knowledge, insight and a deepunderstanding of physics, and more importantly the way to conduct scientificresearch. I am very grateful that he has been always available to discuss themany problems and questions I brought him. I would also like to thank himfor critically reviewing this thesis.
I am very grateful to my second advisor, Professor Helmut Rauch, whowarmly recommended and supported my application for the Austrian Chan-cellor Fellowship from the Austrian Academic Exchange Service, which en-abled my continuation of physics study in Austria.
I would like to express my deep appreciation to my former and presentcolleagues in the Quantum Optics and Foundations of Physics research group.Special thanks go to my friend and permanent colleague, Dr. Dik Bouwmeester,who introduced me to the subject of quantum optics and taught me manydetails of the experiment; he also impressed me with his devotion and ini-tiative. I also especially thank Professor Harald Weinfurter and MatthewDaniell, with whom I collaborated on most of the work. Matthew also hascarefully read through some of the chapters in this thesis. Thanks also tothe other colleagues in the photon laboratory, Dr. Birgit Dopfer, Dr. KlausMattle, Dr. Markus Michler, Dr. Michael Reck, Dr. Surasak Chiangga,Dr. Gregor Weihs, Thomas Jennewein, Alois Mair, Markus Oberparleitnerand Christoph Simon. To the people in the atom laboratory, Professor JorgSchmiedmayer, Dr. Markus Arndt, Dr. Stefan Bernet, Dr. Johannes Den-schlag, Dr. Sonja Frank, Donatella Cassettari, Claudia Keller, Olaf Nairz,and Gerbrand von der Zouw, whose instruments I sometimes stole. And toMrs. Christine Obmascher, Professor Zeilinger’s secretary, for her quiet effi-ciency in the office, and for her continuous help on various matters throughout
3
the years.
Many helpful discussions and much of my knowledge about Bell’s in-equalities are due to my friend and colleague Professor Marek Zukowski, thepermanent visitor to our group.
Dr. Ramon Risco Delgado and Bjorn Hessmo are always remembered. Ienjoyed our discussions about physics, the meaning of life, and all the rest,especially the delicious fish cooked by Bjorn.
This work would have been impossible without the patience and under-standing of my wife Xiao-qing, who supported me during the ups and downsthat are inevitable in such a major undertaking. My parents have providedme with invaluable support through my entire life and education. They haveencouraged and supported me both for starting my undergraduate study ina distant city, and for continuing my doctorate study in another country faraway from my homeland.
Among the many very good teachers I met throughout my academic ca-reer, I especially thank Professor Yong-de Zhang, my undergraduate andgraduate advisor, for his outstanding guidance and continuous concern aboutmy career.
I gratefully acknowledge the support of the Austrian Academic ExchangeService during my study. The financial support of the research in this thesiswas partially from the Austrian Fonds zur Forderung der WissenschaftlichenForschung who with the Schwerpunkt Quantenoptik (Project No. S06502),Project No. F1506 and the TMR-Network ”The Physics of Quantum Infor-mation” of the European Commission.
The attentive reader might notice that a number of text paragraphs weretaken from joint papers of our group because the formulations found thereare difficult to improve.
Finally I sincerely thank all my friends, from all over the world, who mademy years in Innsbruck and Vienna so delightful.
Chapter 1
Introduction
Superposition, one of the most distinct features of the quantum theory, has
been demonstrated in numerous particle analogs of Young’s classic double-
slit interference experiment, such as in electron interferometer [Marton et al.,
1954], neutron interferometer [Rauch et al., 1974] and atom interferometer
[Carnal and Mlynek, 1991; Keith et al., 1991]. However, in multiparticle
systems the superposition principle yields phenomena that are much richer
and more interesting than anything that can be seen in one-particle systems.
Quantum Entanglement, a simple name for superposition in a multipar-
ticle system, was first noticed by Schrodinger [Schrodinger, 1935] and since
then it has baffled generations of physicists. It is at the heart of the dis-
cussions of the Einstein-Podolsky-Rosen (EPR) paradox, of Bell’s inequality,
and of the non-locality of quantum mechanics [Einstein et al., 1935; Bell,
1964]. In recent years, entanglement has become a new focus of activity in
quantum physics because of immense theoretical and experimental progress
both in the foundation of quantum mechanics and in the new field of quantum
information science.
On the theoretical side, while the discovery of the conflict with local
realism following from Greenberger-Horne-Zeilinger (GHZ) entanglement of
three- or more particles [Greenberger et al., 1989; 1990] allows us to per-
5
6 CHAPTER 1. INTRODUCTION
form novel and completely new tests of local realism without inequalities,
the resource of entanglement has also many useful applications in quantum
information processing [Bennett, 1995], including quantum computation and
quantum communication. On the one hand, quantum computation, based on
a controlled manipulation of entangled states of quantum bits, might allow
us to build a new generation of computers which promise to be more pow-
erful than their classical counterparts [Deutsch, 1985], for example, Shor’s
discovery of quantum algorithms [Shor, 1994] enables us to factorize large
integers exponentially faster than the best known classical algorithms. On
the other hand, quantum communication schemes, such as quantum cryp-
tography [Bennett et al, 1992a], dense coding [Bennett and Wiesner, 1992]
and teleportation [Bennett et al., 1993], offer more efficient and secure ways
for the exchange of information in a network.
On the experimental side, the current technology is beginning to allow
us to manipulate rather than just observe individual quantum phenomena.
This opens up the possibility of realizing these above proposals in real experi-
ments. However, although there is fast progress in the theoretical description
of quantum information processing, the difficulties in handling quantum sys-
tems have not yet allowed an equal advance in the experimental realization of
the new proposals. Besides the promising developments of quantum cryptog-
raphy (the first provably secure way to send secret messages), peoples have
only recently succeeded in demonstrating the possibility of quantum dense
coding [Mattle et al., 1996], a way to quantum mechanically enhance data
compression. The main reason for this slow experimental progress is that,
although there exist methods to produce pairs of entangled photons [Kwiat
et al., 1995], entanglement has been demonstrated for atoms [Hagley et al.,
1997] only very recently and it has not been possible thus far to produce en-
tangled states of more than two quanta. Yet, all known methods of quantum
computation are applications of entanglement. The present experimental
challenge is therefore not to build a full-fledged universal quantum computer
straight away but rather to progress from experiments in which we merely
observe quantum interference and entanglement to experiments in which we
7
can control those quantum phenomena in the required way. All this above
leads to the main motivation of our experimental efforts in this dissertation.
Late in 1997, our group successfully achieved the first experimental demon-
stration of quantum teleportation [Bouwmeester, Pan et al., 1997] in which
we disembodied the polarization state of a photon into classical data and
EPR correlations, and then used these ingredients to reincarnate the state
in another photon which has never been anywhere near the first photon.
Due to our first realization of quantum teleportation, experimental research
in the field of quantum information is now attracting increasing attention
from both academia and industry. In the year of 1998, several groups in the
world achieved a series of important advances involving quantum computa-
tion and teleportation: In February, De Martini’s group [Boschi et al., 1998]
reported an optical realization for Popescu’s scheme–a variant of the original
teleportation proposal. In May, we [Pan et al., 1998b] experimentally real-
ized entanglement swapping, that is, teleportation of completely undefined
quantum state; Chuang and his coworkers, meanwhile, reported the first ex-
perimental realization of the Deutsch-Jozsa quantum algorithm using a bulk
nuclear magnetic resonance (NMR) technique [Chuang et al., 1998]. Then
in October, Kimble’s group [Furusawa et al., 1998] succeeded in teleporting
information on the amplitude and phase of an entire light beam to another
beam. In November, Nielsen et al. [Nielsen et al., 1998] teleported quan-
tum information from the nucleus of carbon atom to that of a neighboring
hydrogen atom.
Recently, according to the proposal for production of GHZ entanglement
out of entangled pairs [Zeilinger et al., 1997], we implemented a source of GHZ
entanglement for three spatially separated photons [Bouwmeester, Pan et al.,
1999], which is a further development of the technique that has been used
in our previous experiments for teleportation and entanglement swapping.
Such a source, for the first time, opens the door to demonstrate the GHZ
theorem. The first three-particle test of local realism without inequalities has
been done most recently [Pan et al., 1999a]. All these significant advances
8 CHAPTER 1. INTRODUCTION
greatly promote experimental research both in the foundations of quantum
mechanics and in the field of quantum information.
The aim of the thesis is to report the first experimental realization of
teleportation for arbitrary quantum states, to report the first observation of
GHZ entanglement for three spatially separated photons, and to report the
first demonstration of non-locality of quantum mechanics for nonstatistical
predictions of the theory. The main contents of the dissertation are organized
as follow:
Chapter 1 is a concise introduction to the applications of quantum entan-
glement. We briefly review the current theoretical and experimental advances
both in the foundation of quantum mechanics and in the new field of quantum
information science.
As all applications of entanglement necessitate both preparation and mea-
surement of entangled states, in Chapter 2, we shall theoretically describe
how quantum networks with simple quantum logic gates and a handful of
qubits allow us to control and manipulate quantum entanglement. Due to
the present absence of a general quantum gate, meanwhile we also present
a practical way to generate and identify entangled states, which constitutes
the basis of all experiments in the dissertation.
Quantum teleportation, the transmission and reconstruction over arbi-
trary distances of the state of a quantum system, is experimentally demon-
strated in Chapter 3. We describe in detail the theoretical and experimental
schemes of quantum teleportation. During teleportation, an initial photon
which carries the polarization that is to be transferred and one of a pair
of entangled photons are subjected to a measurement such that the second
photon of the entangled pair acquires the polarization of the initial pho-
ton. Quantum teleportation will be a critical ingredient for future quantum
computation networks.
Entanglement swapping enables one to entangle particles that never phys-
9
ically interacted with one another or which have never been dynamically cou-
pled by any other means. Chapter 4 reports in detail an optical experimental
realization of entanglement swapping. As we will see below, entanglement
swapping, besides its interest to fundamental physics, will have a number of
important applications in quantum communication.
In Chapter 5, we report the first observation of polarization entangle-
ment of three spatially separated photons. Such an entangled state is the
long-coveted GHZ state. In addition to facilitating more advanced forms of
quantum cryptography, our GHZ state will help provide a non-statistical test
of the foundations of quantum physics.
Chapter 6 is concerned with the test of local realism via a GHZ state.
Though previous experiments, based on observation of the entangled state of
two photons, have provided highly convincing evidence against local realism,
these ”Bell’s inequalities” tests require the measurement of many pairs of
entangled photons to build up a body of statistical evidence against the
idea. In contrast, the GHZ theorem described in the chapter shows that
in principle studying a single set of properties in the GHZ photons could
verify the predictions of quantum mechanics while contradicting those of
local realism. Using the source exploited in Chapter 5, we present here the
first experimental demonstration of the GHZ theorem.
Chapter 7 involves conclusions and outlooks. In the chapter, we briefly
summarize the results in the thesis, and discuss the differences among those
recent experimental advances of quantum teleportation. Finally, we also give
some prospects for future experiments.
Chapter 2
Manipulation of EntangledStates
It is well known that the preparation and measurement of Bell states is es-
sential for quantum dense coding [Bennett and Wiesner, 1992], for quantum
teleportation [Bennett et al., 1993], and for entanglement swapping [Zukowski
et al., 1993], the extension to the GHZ situation allows us to generalize the
two to the multi-particle case. This chapter consists of three sections. First,
we shall theoretically describe how using only single-quantum-bit (qubit)
operations and controlled-NOT gates [Barenco et al., 1995a; 1995b] one
can construct a suitable quantum network to produce and identify any of
the maximally entangled states for any number of particles [Bruss et al.,
1997]. Second, since until now such quantum networks have not yet been
built in the laboratory, we shall also describe the existing Bell-state analyzer
[Weinfurter, 1994; Braustein and Mann, 1995] and then present a universal
scheme and practically realizable procedures, by which one can readily iden-
tify two of the maximally entangled states of any number of photons [Pan
and Zeilinger, 1998]. At the end of this chapter a high intensity source of
polarization-entangled photon pairs [Kwiat et al., 1995], which has been used
in all experiments of the dissertation, is described, and some basic alignment
procedures are discussed briefly.
10
2.1. QUANTUM NETWORK AND ITS APPLICATIONS 11
Figure 2.1: Graphical representations of Hadamard and the quantumcontrolled-NOT gates. Here, a + b denotes addition modulo 2.
2.1 Quantum network and its applications
Generally, a bit is a classical system with two Boolean states 0 and 1, while
a qubit means a generic two-state quantum system with a chosen ”computa-
tional basis” {|0〉, |1〉} (e.g. the polarization of a photon or a spin-12particle).
A quantum logic gate is an elementary device which performs a fixed unitary
operation on selected qubits in a fixed period of time. Single-qubit quantum
logic gates are rather trivial and can be implemented, for example, by ex-
citing selected atomic transitions with laser pulses of controllable frequency,
intensity and duration. In fact, using a simple Hadamard gate and the quan-
tum controlled-NOT gates, one can prepare and identify any of N -particle
entangled states. The action of a Hadamard gate (Fig. 2.1a) is equivalent
to the following unitary transformation:
|0〉 → 1√2(|0〉+ |1〉)
|1〉 → 1√2(|0〉 − |1〉) (2.1)
12 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
and the controlled-NOT gate(Fig. 2.1b) flips the second of two qubits if and
only if the first is |1〉, namely
|0〉|0〉 → |0〉|0〉|0〉|1〉 → |0〉|1〉|1〉|0〉 → |1〉|1〉|1〉|1〉 → |1〉|0〉
(2.2)
Consider now the network shown in Figure 2.2a. Under the action of the
gates on the left-hand side of the network, the input two-particle states will
undergo a series of unitary transformation. For example, if the input state
is |0〉|0〉, after passing through the two gates it will be transformed into:
|0〉|0〉Hadmard−−− −→ 1√
2(|0〉|0〉+ |1〉|0〉)
C−NOT 12−−− −→ 1√
2(|0〉|0〉|+ |1〉|1〉)
(2.3)
which is one of the four maximally entangled Bell states,
|Ψ±〉 = 1√2(|0〉 |1〉 ± |1〉 |0〉)
|Φ±〉 = 1√2(|0〉 |0〉 ± |1〉 |1〉) .
(2.4)
Correspondingly, the network could also prepare the two qubits in one of the
remaining three Bell states:
|1〉|0〉 −→ 1√2(|0〉|0〉| − |1〉|1〉) (2.5)
|0〉|1〉 −→ 1√2(|0〉|1〉|+ |1〉|0〉) (2.6)
|1〉|1〉 −→ 1√2(|0〉|1〉| − |1〉|0〉) (2.7)
It is easy to verify that the reversed quantum network (right-hand side of
Figure 2.2a) can be used to implement the so-called Bell measurement on the
2.1. QUANTUM NETWORK AND ITS APPLICATIONS 13
Figure 2.2: (a) The Bell measurement: the gates on the left hand side allowus to generate the four Bell states from the four possible different inputs.Reversing the order of the gates (right-hand side of the diagram) correspondsto a Bell measurement. (b) GHZ measurement: the same as in (a) for theeight GHZ states.
14 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
two qubits by disentangling the Bell states. In this way, the Bell measurement
is reduced to two single-particle measurements. The method can be directly
extended to a three-qubit case. Figure 2.2b shows how to prepare eight maxi-
mally entangled three-particle states, known as the GHZ states [Greenberger
1989,1990]. Reversing the procedure we obtain the unitary transformation
which reduces the GHZ measurement to the three single-particle measure-
ments.
We can write this in the following compact form, where a and b can each
take the values 0 and 1 and a and b denote NOT-a and NOT-b, respectively:
|0〉|a〉|b〉 ⇐⇒ 1√2
(|0〉|a〉||b〉+ |1〉|a〉|b〉
)(2.8)
|1〉|a〉|b〉 ⇐⇒ 1√2
(|0〉|a〉||b〉 − |1〉|a〉|b〉
)(2.9)
Let us also mention that the GHZ measurement provides an interesting
possibility of labeling the GHZ states via the corresponding binary output.
The three output bits then have the following meanings:
(i) The first output bit tells us whether the number of |0′〉 in the GHZ
state, written in the conjugate basis (this is given by |0′〉 = (1/√2)(|0〉+ |1〉)
and |1′〉 = (1/√2)(|0〉 − |1〉)), is even or odd. If the first output bit is |0〉,
there is an odd number of |0′〉s in the conjugate basis, otherwise an even
number.
(ii) The second output bit indicates whether the first two bits in the GHZ
superposition are the same or different. If the second output bit is |0〉, theyare the same.
(iii) The third output bit provides the same information with respect to
the first and third bit of the GHZ superposition.
Finally, we would like to emphasize that, in general, any measurement on
any number of qubits can be implemented using only single-qubit operations
2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 15
Figure 2.3: The beam splitter coherently transforms two input spatial modes(a, b) into two output spatial modes (c, d)
and the quantum controlled-NOT gates.
This follows from the fact that the quantum controlled-NOT gate, to-
gether with relatively trivial single-qubit operations, forms an adequate set
of quantum gates, i.e., the set from which any unitary operation may be
built [Barenco et al., 1995a; 1995b]. Thus if we want to measure observable
A pertaining to n qubits, we could construct a compensating unitary trans-
formation U which maps 2n states of the form |a1〉|a2〉...|an〉, where ai = 0, 1,
into the eigenstates of A. This allows both to prepare the eigenstates of A,
which in general can be highly entangled, and to reduce the measurement
described by A to n simple, single-qubit measurements.
2.2 Practical schemes for entangled-state anal-
ysis
Though quantum networks present a novel way to manipulate entangled
states, their experimental realization remains to be a challenge for the future
due to the lack of a quantum controlled-NOT gate. However, while no com-
plete Bell-state measurement procedure exists, we can already experimentally
identify two of the maximally entangled states of N photons.
16 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
2.2.1 Bell-state analysis
The Bell-state analyzer first suggested by Weinfurter et al. is based on the
two-photon interference effect at a 50:50 standard beam splitter. The beam
splitter has two spatial input modes a and b and two output modes c and
d (Fig. 2.3). Quantum mechanically, the action of the beam splitter on the
input modes can be written as
|a〉 −→ i√2|c〉+ 1√
2|d〉
|b〉 −→ 1√2|c〉+ i√
2|d〉 (2.10)
where, e.g. |a〉 describes the spatial quantum state of the particle in input
beam a. Eq. 2.10 describes the fact that the particle can be found with equal
probability (50%) in either of the output modes c and d, no matter through
which input beam it came. Here the factor i in Eq. 2.10 is a consequence of
unitarity. It corresponds physically to a phase jump upon reflection at the
semi-transparent mirror [Zeilinger, 1981]. Note that a standard beam splitter
is polarization independent, and thus has no effect on the polarization state
of the photon.
Let us now consider our beam splitter with two incident photons, 1 and
2, photon 1 in input beam a, and photon 2 in input beam b. Suppose that
photon 1 is in polarization state α|H〉1+β|V 〉1, and photon 2 is in polarizationstate γ|H〉2 + δ|V 〉2 ( here H and V denote horizontal and vertical linear
polarizations, and |α|2 + |β|2 = 1, |γ|2 + |δ|2 = 1). They each have the same
probability p = 0.5 to transmit the beam splitter or be reflected. Thus, four
different possibilities arise (Fig. 2.4).
(1) Both particles are reflected, (2) both particles are transmitted, (3) the
upper particle is reflected, the lower one is transmitted, and (4) the upper one
is transmitted and the lower one is reflected. Each of the four occurs with the
same probability, and one has to investigate now whether any interference
between these processes is possible. For distinguishable particles, for example
for classical ones, no interference arises and we thus arrive at the prediction
2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 17
Figure 2.4: Two particles incident onto a beam splitter, one from each side.Four possibilities exist how the two particles can leave the beam splitter.
18 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
that in two of the cases, that is, with total probability p = 0.5, the two
particles end up in different output ports and, with probability p = 0.25, both
particles end up in the upper output beam and, with the same probability
p = 0.25, they end up in the lower output beam.
Let us now assume that the two photons have the same frequency and
arrive at the beam splitter simultaneously. As a result they are quantum
mechanically indistinguishable. In this case it is not possible, not even in
principle, to decide which of the incident particles ended up in a given output
port, we therefore have to consider coherent superpositions of the amplitudes
for these different possibilities. To show how the Bell-state analyzer works,
consider the input state
|ψi〉 = (α|H〉1 + β|V 〉1)|a〉1·(γ|H〉2 + δ|V 〉2)|b〉2.
(2.11)
where, for example, the first term in the equation indicates photon 1 with a
polarization state α|H〉1 + β|V 〉1 is in input mode a.
As shown in Eq. 2.10, for photons 1 and 2 passing through the beam split-
ter their spatial modes will undergo a corresponding unitary transformation.
The state in Eq. 2.11 thus evolves into
|ψf〉12 = 1√2(α|H〉1 + β|V 〉1)(i|c〉1 + |d〉1)·
1√2(γ|H〉2 + δ|V 〉2)(|c〉2 + i|d〉2).
(2.12)
It should be noted that photons 1 and 2 are not distinguishable anymore
after passing through the beam splitter. The total two-photon state including
both the spatial and the spin part, therefore, has to obey bosonic quantum
statistics. This implies that the outgoing physical state must be symmetric
under exchange of labels 1 and 2. To do so, one should symmetrize the state
|ψf〉12, that is, also include its exchange wave-function
|ψf〉21 = 1√2(α|H〉2 + β|V 〉2)(i|c〉2 + |d〉2)·
1√2(γ|H〉1 + δ|V 〉1)(|c〉1 + i|d〉1).
(2.13)
2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 19
The final outgoing state therefore reads
|ψf〉 =1√2(|ψf〉12 + |ψf〉21), (2.14)
and consequently we have
|ψf〉 = 12√
2[(αγ + βδ)(|H〉1|H〉2 + |V 〉1|V 〉2) · i(|c〉1|c〉2 + |d〉1|d〉2)+(αγ − βδ)(|H〉1|H〉2 − |V 〉1|V 〉2) · i(|c〉1|c〉2 + |d〉1|d〉2)+(αδ + βγ)(|H〉1|V 〉2 + |V 〉1|H〉2) · i(|c〉1|c〉2 + |d〉1|d〉2)+(αδ − βγ)(|H〉1|V 〉2 − |V 〉1|H〉2) · (|d〉1|c〉2 − |c〉1|d〉2)].
(2.15)
As we will see below, Eq. 2.15 allows us to readily project the two-photon
state into two of the four maximally polarization-entangled states:
|Ψ±〉12 = 1√2(|H〉1 |V〉2 ± |V〉1 |H〉2)
|Φ±〉12 = 1√2(|H〉1 |H〉2 ± |V〉1 |V〉2) .
(2.16)
From Eq. 2.15, it is easy to verify that the two photons proceed after the
beam splitter in different emerging beams if, and only if, their polarization
state is in the state |Ψ−〉12 (refer to the fourth term of Eq. 2.15). Thus
we arrived at a possibility to identify one of the four Bell states, |Ψ−〉12,
uniquely on the basis that it is the only one which gives rise to a detection
of one photon in each of the outgoing beams of the beam splitter. For a
full analysis, we further need a way to distinguish between the other three
states, |Ψ+〉12, |Φ+〉12, and |Φ−〉12. Again, one can easily find that it is only in
the state |Ψ+〉12 that the two emerging photons have different polarizations.
In the two |Φ±〉12 states, they always share the same polarization. Thus,
a further step in Bell-state analysis implies that one inserts a two-channel
polarizer into each of the output ports of the beam splitter. Then, only the
state |Ψ+〉12 will give a coincidence count between the two output ports of
the polarizer on either side of the beam splitter. Yet, both |Φ±〉12 states will
give rise to the same joint detection of the two photons in either detector
after the final polarizer. By making use of two-particle interference effects
at a beam splitter, we can thus distinguish two of the four Bell states via
two-fold coincidence analysis.
20 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
2.2.2 GHZ-state analyzer
As an application of the concept of quantum erasure, we present in this part a
practical GHZ-state analyzer for identifying two of the N -particle entangled
states [Pan 1998a]. The basic elements of the experimental setup are just
polarizing beam splitters(PBS) and half-wave plates(HWP).
Before starting to discuss the GHZ-state analyzer, let’s first give a mod-
ified version of the Bell-state analyzer, which is similar to but different from
the former one shown in Fig. 2.3. Consider the arrangement of Fig. 2.5.
Two identical photons enter our Bell-state analyzer from modes A and B re-
spectively. Suppose they are in the most general polarization-superposition
state
|ψin〉 = α|HA〉|HB〉+ β|HA〉|VB〉+ γ|VA〉|HB〉+ δ|VA〉|VB〉 (2.17)
Because the polarizing beam splitter PBS transmits only the horizontal polar-
ization component and reflects the vertical component, after passing through
PBSAB the incident state will evolve into
|ψin〉 −→ α|HA2〉|HB1〉+ β|HA2〉|VB2〉+ γ|VA1〉|HB1〉+ δ|VA1〉|VB2〉 (2.18)
Where subscript ij(i = A,B, j = 1, 2) denotes the transformation from in-
put mode i to output mode j. Suppose that these two photons A and B
arrive at the polarizing beam splitter PBSAB simultaneously, and therefore
their spatial wavefuctions overlap each other. Then, according to the indis-
tinguishability of identical particles, we can directly denote HA2 as H2, VA1
as V1, HB1 as H1, and VB2 as V2. Thus, Eq. 2.18 reads
α|H1〉|H2〉+ β|H2〉|V2〉+ γ|V1〉|H1〉+ δ|V1〉|V2〉 (2.19)
Note that for the terms |H1〉|H2〉 and |V1〉|V2〉 the two photons are in
different output ports, and while for the terms |H2〉|V2〉 and |V1〉|H1〉 both
2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 21
Figure 2.5: A modified Bell-state analyzer. Two indistinguishable photonsenter the Bell-state analyzer from input ports A and B. PBSAB, PBS1 andPBS2 are three polarizing beam splitters, which transmit the horizontal polar-ization component and reflect the vertical component. DH1, DV 1, DH2, andDV 2 are four photon-counting detectors.
22 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
photons are in the same output port. Therefore, we can identify states |ψd〉 =α|H1〉|H2〉+δ|V1〉|V2〉 and |ψs〉 = β|H2〉|V2〉+γ|H1〉|V1〉 using the coincidencebetween detectors in mode 1 and in mode 2. Obviously, in terms of the Bell
states, ψd can be rewritten as
|ψd〉 =1√2(α+ δ)|Φ+〉12 +
1√2(α− δ)|Φ−〉12 (2.20)
Thus, in order to finish the Bell-state measurement we now only need to
identify states |Φ+〉12 and |Φ−〉12, that is , we have to determine the relative
phase between terms of |H1〉|H2〉 and |V1〉|V2〉. Let the angle between the
HWP axis and the horizontal direction be 22.50 such that it corresponds to
a 450 rotation of the polarization. Therefore, for a photon passing the HWP,
its polarization state will undergo the following unitary transformation:
|Hi〉 −→ 1√2(|Hi〉+ |Vi〉)
|Vi〉 −→ 1√2(|Hi〉 − |Vi〉)
where i = 1, 2. Finally, |Φ+〉12 and |Φ−〉12 will thus be transformed into
|Φ+〉12 → |Φ+〉12 =1√2(|H1〉|H2〉+ |V1〉|V2〉) (2.21)
|Φ−〉12 → |Ψ+〉12 =1√2(|H1〉|V2〉+ |V1〉|H2〉) (2.22)
The above analysis show that we can easily identify two of the four incident
Bell states. Specifically, if we observe a coincidence either between detectors
DH1 and DH2 or DV 1 and DV 2, then the incident state was 1√2(|HA〉|HB〉 +
|VA〉|VB〉). On the other hand if we observe coincidence between detectors
DH1 and DV 2 or DV 1 and DH2, then the incident state was1√2(|HA〉|HB〉 −
|VA〉|VB〉). The other two incident Bell states will lead to no coincidence
between detectors in mode 1 and in mode 2. Such states are signified by
some kind of superposition of |HA〉|VB〉 and |VA〉|HB〉. This concludes our
demonstration that we can identify two of the four Bell states using the
coincidence between modes 1 and 2.
2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 23
Figure 2.6: A GHZ-state analyzer. Three photons incident one each in modesA, B, and C will give rise to distinct 3-fold coincidence if they are in the GHZ-states |Φ+〉 or |Φ−〉. All the notations are the same as those in Fig. 2.5.
The reason why we discuss the modified version of Bell-state analyzer is
that the above scheme can directly be generalized to the N -particle case.
Making use of its basic idea, one can easily construct a type of GHZ-state
analyzer by which one can immediately identify two of the 2N maximally
entangled GHZ states.
For example, in the case of three identical photons, the eight maximally
entangled GHZ states are given by
|Φ±〉 = 1√2(|H〉|H〉|H〉 ± |V 〉|V 〉|V 〉) (2.23)
|Ψ1±〉 = 1√
2(|V 〉|H〉|H〉 ± |H〉|V 〉|V 〉) (2.24)
24 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
|Ψ2±〉 = 1√
2(|H〉|V 〉|H〉 ± |V 〉|H〉|V 〉) (2.25)
|Ψ3±〉 = 1√
2(|H〉|H〉|V 〉 ± |V 〉|V 〉|H〉) (2.26)
Where |Ψi±〉 designates that GHZ-state where the polarization of photon
i is different from the other two. Consider now the setup of Fig. 2.6 and
suppose that three photons enter the GHZ analyzer, each one from modes A,
B and C respectively. A suitable arrangement can be realized such that the
photon coming from mode A and the one coming from mode B overlap at
PBSAB and, thus they are correspondingly transformed into mode 1 and into
mode BC. Let’s further suppose that the photons from mode BC and mode
C overlap each other at PBSBC . Thus, following the above demonstration
for the case of the Bell-state analyzer, it is easy to find that the eight GHZ
states above will correspondingly evolve into
1√2(|H1〉|H2〉|H3〉 ± |V1〉|V2〉|V3〉) (2.27)
1√2(|H1〉|H3〉|V3〉 ± |V1〉|V2〉|H2〉) (2.28)
1√2(|H2〉|V2〉|H3〉 ± |H1〉|V1〉|V3〉) (2.29)
1√2(|H1〉|V1〉|H2〉 ± |V2〉|H3〉|V3〉) (2.30)
immediately after these three photons passed through PBSAB and PBSBC ,
and before they enter the half-wave plates(HWP). Here, e.g. Hi denotes a
photon with polarization H in output mode i.
From Eqs. 2.27-2.30, it is evident that one can observe three-fold coinci-
dence between modes 1, 2 and 3 only for the state of Eq. 2.27. For the other
states, there are always two particles in the same mode. We can thus dis-
tinguish the two states 1√2(|H〉|H〉|H〉± |V 〉|V 〉|V 〉) from the other six GHZ
states. Furthermore, after the states of Eq. 2.27 pass through the HWP, we
finally obtain
2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 25
1√2(|H1〉|H2〉|H3〉+ |V1〉|V2〉|V3〉)→ 1
2(|H1〉|H2〉|H3〉+ |H1〉|V2〉|V3〉+|V1〉|H2〉|V3〉+ |V1〉|V2〉|H3〉)
(2.31)
1√2(|H1〉|H2〉|H3〉 − |V1〉|V2〉|V3〉)→ 1
2(|H1〉|H2〉|V3〉+ |H1〉|V2〉|H3〉+|V1〉|H2〉|H3〉+ |V1〉|V2〉|V3〉)
(2.32)
Thus, using three-fold coincidence we can readily identify the relative phase
between states |H〉|H〉|H〉 and |V 〉|V 〉|V 〉. This is because only the initial
state |Φ+〉 leads to coincidence between detectors DH1, DH2 and DH3 (or
H1V2V3, V1H2V3, V1V2H3). On the other hand, only the state |Φ−〉 leadsto coincidence between detectors DH1, DH2 and DV 3 (or H1V2H3, V1H2H3,
V1V2V3). Or, in conclusion, states |Φ+〉 and |Φ−〉 are identified by coinci-
dences between all three output modes 1, 2, and 3. They can be distinguished
because behind the half wave plates |Φ+〉 results in one or three horizontally,and zero or two vertically polarized photons, while |Φ−〉 results in zero or twohorizontally polarized photons and one or three vertically polarized ones.
Our GHZ-state analyzer has many possible applications. For example
the three photons entering via the modes A, B and C respectively could each
come from one entangled pair. Then projection of these three photons using
the GHZ-state analyzer onto the GHZ state Φ+ or Φ− implies that the other
three photons emerging from each pair will be prepared in a GHZ state.
It is clear that our scheme can readily be generalized to analyze entangled
states consisting of more than three photons by just adding more polarizing
beam splitters and half wave plates. Also, identification of analogs of our
scheme for GHZ-state analysis of atoms or of mode entangled states instead
of polarization entangled ones is straightforward.
Here, it is worth noting that an extension of the above scheme would pro-
vide us with a conditional GHZ-state source by which we can conveniently
observe four-particle GHZ correlations and further prepare three freely propa-
gating particles in a GHZ state [Zeilinger et al, 1997; Pan and Zeilinger, 1998;
26 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
Figure 2.7: A three- and four-photon polarization-entanglement source. Thephoton sources, A and B, pumped by short pulses, each one emits a photonpair in the superposition HH+V V . Then, one photon coming from source Aand one coming from source B overlap at the polarizing beam splitter PBS1.F is a narrow filter, PBS is a special polarizing beam splitter which transmits45◦ polarization and reflects −45◦ polarization. DT1 and DT2 are two single-photon detectors.
2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 27
Pan and Zeilinger, 1999b]. Consider two sources A and B (see Fig. 2.7) each
one emitting a photon pair. Consider for simplicity that the photons emitted
by sources A and B are both in the same entangled state 1√2(|H〉|H〉+|V 〉|V 〉).
Then, using very analogous arguments as above we find that the state of the
four particles immediately after passage through the polarizing beam splitter
PBS1 will be the superposition
12(|H1〉|H2〉|H3〉|H4〉+|V1〉|V2〉|V3〉|V4〉+|H1〉|H3〉|V3〉|V4〉+|V1〉|V2〉|H2〉|H4〉)
(2.33)
Again, only for the superposition |H1〉|H2〉|H3〉|H4〉 + |V1〉|V2〉|V3〉|V4〉 wewould observe four-fold coincidence. Therefore, we then know these four par-
ticles are in the superposition |H1〉|H2〉|H3〉|H4〉+ |V1〉|V2〉|V3〉|V4〉 as soon aswe observe four-fold coincidence. Note that here the GHZ state is not directly
prepared but we know that the four particles are in a GHZ state under the
condition that one particle each is detected in each of the outgoing beams 1,
2, 3 and 4. This is a much weaker condition than any post-selection proce-
dure which might be based on properties of the particles. In an experiment
our case will not be distinguishable from the real situation occurring anyway
because of finite detector efficiency. That is, from a practical point of view,
even if one definitely prepares a full GHZ state one only will observe four-
fold coincidence in a fraction of time anyway. Thus, we conclude that using
our conditional GHZ-state one will be able to experimentally demonstrate
all features of a 4-particle GHZ state.
Yet, in the meantime we would like to note that our scheme in Fig. 2.7
also allows us to generate unconditional three-particle GHZ states via so-
called entangled entanglement [Krenn and Zeilinger, 1996]. For example,
one could analyze the polarization state of photon 2 by passing it through a
polarizing beam splitter PBS selecting 45◦ and −45◦) polarization. Then thepolarization state of the remaining three photons 1, 3 and 4 will be projected
28 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
into
1√2(|H1〉|H3〉|H4〉+ |V1〉|V3〉|V4〉
if and only if detector DT1 detects a single photon. Correspondingly, the
state of photons 1, 3 and 4 will be projected into
1√2(|H1〉|H3〉|H4〉 − |V1〉|V3〉|V4〉
if and only if detector DT2 detects a single photon. In the scheme, the
detection of photon 2 actually plays the double role of both getting rid of the
last two terms in Eq. 2.33 and projecting the remaining three photons into
a spatially separated and freely propagating GHZ state. Such a GHZ-state
could be extremely useful both in further test of local realism versus quantum
mechanics and in future application of third-man cryptography.
Finally we would like to note that in all these schemes we have used the
principle of quantum erasure in a way that behind our GHZ-state analyzer at
least some of the photons registered cannot be identified anymore as to which
source they came from. This implies very specific experimental schemes, be-
cause the particles might have been created at different times. One theo-
retical possibility is to apply the principle of ultra-coincidence [Zukowski et
al., 1993]. This means that the photons must be registered within a time
short compared to their coherence time. For practical reasons, i.e. the un-
availability of sufficiently fast detectors, the scheme cannot be realized at
present. Yet, alternatively, one can create the particles within a time inter-
val small compared to their coherence times. This in practice implies the use
of pulsed sources and of filters behind them which introduce coherence times
larger than the pulse length [Zukowski et al., 1995]. Such a scheme has been
successfully used in our following experiments on quantum teleportation, en-
tanglement swapping and three-particle GHZ entanglement.
2.3. POLARIZATION-ENTANGLED PHOTON PAIRS 29
2.3 Polarization-entangled photon pairs
Because of the same reason, i.e. the absence of a quantum logic gate, at the
moment there are not so many possibilities to create states of entangled par-
ticles in the laboratory. Fortunately, the process of spontaneous parametric
down-conversion provides mechanisms by which pairs of entangled photons
can be produced with reasonable intensity and in good purity. In the down-
conversion process, one uses a non-centrolsymmetric crystal with nonlinear
electric susceptibility. In such a medium, an incoming photon can decay
with relatively small probability into two photons in a way that energy and
momentum inside the crystal are conserved.
Here we will describe a simple technique to produce polarization-entangled
photon pairs using the process of noncollinear type-II parametric down-
conversion [Kwiat et al., 1995]. In the experiment, the desired polarization-
entangled state is produced directly out of a single nonlinear crystal [BBO
(beta-barium-borate)]. In that process, the two photons are emitted with dif-
ferent polarizations(Fig. 2.8). Calculating the emission direction of the pho-
tons [Caro and Garuccio, 1994; Kwiat et al, 1995], one notices that photons
of each polarization are emitted into one cone in such a way that momenta of
two photons always add up to the momentum of the pump photon. Thus, the
emission direction of each individual photon is completely uncertain within
the cone, but once one photon is registered, and thus its emission direction
is defined, the other photon is found just exactly opposite from the pump
beam on the other cone. The total quantum mechanical state is therefore
extremely rich and is a superposition of all such pairs of emission modes.
The interesting point is now that the crystal can be cut and arranged
such that the two cones intersect, as shown in Fig. 2.8. Then, along the lines
of intersection, the polarization of neither photon is defined, but what is de-
fined is the fact that the two photons have to have different polarizations.
This contains all the necessary features of entanglement in a nutshell. Mea-
surement on each of the photons separately is totally random and gives with
30 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
Figure 2.8: Principle of type-II parametric down-conversion. Inside a nonlin-ear crystal(here, BBO), an incoming pump photon can decay spontaneouslyinto two photons. Two down-converted photons arise polarized orthogonallyto each other. Each photon is emitted into a cone, and the photon on the topcone is vertically polarized while its exactly opposite partner in the bottomcone is horizontally polarized. Along the directions where the two cones in-tersect, their polarizations are undefined; all that is known is that they haveto be different, which results in polarization entanglement between the twophotons in beams A and B.
2.3. POLARIZATION-ENTANGLED PHOTON PAIRS 31
equal probability vertical or horizontal polarization. But once one photon,
for example photon A, is measured, the polarization of the other photon B
is orthogonal! Choosing an appropriate basis, e.g. |H〉 and |V 〉, the stateemerging through the two beams A and B thus is a superposition of |H〉|V 〉and |V 〉|H〉, say
1√2(|H〉A|V 〉B + eiα|V 〉A|H〉B) (2.34)
where the relative phase α arises from the crystal birefringence, and an overall
phase shift is omitted.
Using an additional birefringent phase shifter (or even slightly rotating
the down-conversion crystal itself), the value of α can be set as desired, e.g.,
to the values 0 or π. Somewhat surprisingly, a net phase shift of π may
be obtained by a 90◦ rotation of a quarter wave plate in one of the paths.
Similarly, a half wave plate in one path can be used to change horizontal
polarization to vertical and vice versa. One can thus very easily produce any
of the four EPR-Bell states in Eq. 2.16.
The birefringent nature of the down-conversion crystal complicates the
actual entangled state produced, since the ordinary and extraordinary pho-
tons have different velocities inside the crystal, and propagate along different
directions even though they become collinear outside the crystal (an effect
well known from calcite prisms, for example). The resulting longitudinal and
transverse walk-offs between the two terms in the state (2.34) are maximal
for pairs created near the entrance face, which consequently acquire a relative
time delay δT = L(1/uo − 1/ue) (L is the crystal length, and uo and ue are
the ordinary and extraordinary group velocities, respectively) and a relative
lateral displacement d = L tan ρ (ρ is the angle between the ordinary and
extraordinary beams inside the crystal). If δT ≥ τc, the coherence time of
the down-conversion light, then the terms in Eq. 2.34 become, in principle,
distinguishable by the order in which the detectors would fire, and no inter-
ference will be observable. Similarly, if d is larger than the coherence width,
32 CHAPTER 2. MANIPULATION OF ENTANGLED STATES
the terms can become partially labeled by their spatial location.
Because the photons are produced coherently along the entire length of
the crystal, one can completely compensate for the longitudinal walk-off [Ru-
bin et al., 1994]—after compensation, interference occurs pairwise between
processes where the photon pair is created at distances ±x from the middle
of the crystal. The ideal compensation is therefore to use two crystals, one in
each path, which are identical to the down-conversion crystal, but only half
as long. If the polarization of the light is first rotated by 90◦ (e.g., with a
half wave plate), the retardations of the o and e components are exchanged
and complete temporal indistinguishability is restored (δT = 0). The same
method provides optimal compensation for the transverse walk-off effect as
well. Here, the compensation crystals were oriented along the same direc-
tion as that of the down-conversion crystal. In the following experiments we
always slightly rotate the orientation of one of the compensation crystals to
tune the relative phase α = π.
The BBO crystal used in our experiments is 2.0mm long and was cut at
θpm = 43.5◦ (the angle between the crystal optic axis and the pump) in order
to result in a well-defined the intersection between the two cones. The two
cone-overlap directions, selected by irises before the detectors, were conse-
quently separated by 5◦ when the pump beam is precisely orthogonal to the
surface of the crystal. The transverse walk-off d (0.2mm) was small com-
pared to the coherent pump beam width (2mm), so the associated labeling
effect was minimal. However, it was necessary to compensate for longitudinal
walk-off, since our 2.0mm BBO crystal produced δT = 260fs, while τc [de-
termined by the collection irises and interference filters (centered at 788nm,
4.6nm FWHM)] was about of the same order. As discussed above, we used
an additional BBO crystal (1.0mm thickness, θpm = 43.5◦) in each of the
paths, preceded by a half wave plate to exchange the roles of the horizontal
and vertical polarizations.
Chapter 3
Quantum Teleportation
3.1 Introduction
The dream of teleportation is to be able to travel by simply reappearing at
some distant location. An object to be teleported can be fully characterized
by its properties, which in classical physics can be determined by measure-
ment. To make a copy of that object at a distant location one does not
need the original parts and pieces–all that is needed is to send the scanned
information so that it can be used for reconstructing the object. But how
precisely can this be a true copy of the original? What if these parts and
pieces are electrons, atoms and molecules? What happens to their individual
quantum properties, which according to Heisenberg’s uncertainty principle
cannot be measured with arbitrary precision?
Bennett et al. [Bennett et al., 1993] have suggested that it is possible to
transfer the quantum state of a particle onto another particle– the process of
quantum teleportation–provided one does not get any information about the
state in the course of this transformation. This requirement can be fulfilled
by using entanglement, according to Schrodinger, the essential feature of
quantum mechanics [Schrodinger, 1935]. It describes correlations between
quantum systems much stronger than any classical correlation could be.
33
34 CHAPTER 3. QUANTUM TELEPORTATION
The possibility of transferring quantum information is one of the cor-
nerstones of the emerging field of quantum communication and quantum
computation [Bennett, 1995]. As we will see below, quantum teleportation
is indeed not only a critical ingredient for quantum computation and com-
munication, its experimental realization will also allow new studies of the
fundamentals of quantum theory.
In the present chapter, we report the first experimental verification of
quantum teleportation [Bouwmeester, Pan et al., 1997]. By producing pairs
of entangled photons by the process of parametric down-conversion and using
two-photon interferometry for analyzing entanglement, one could transfer
a quantum property (in our case the polarization state) from one photon
to another. The methods developed for this experiment will be of great
importance both for exploring the field of quantum information as well as
for future experiments on the foundations of quantum mechanics.
3.2 Quantum teleportation–the idea
3.2.1 The problem
To make the problem of transferring quantum information clearer suppose
that Alice has some particle in a certain quantum state |Ψ〉 and she wants
Bob, at a distant location, to have a particle in that state. There is certainly
the possibility to send Bob the particle directly. But suppose that the com-
munication channel between Alice and Bob is not good enough at the time
of the procedure to preserve the necessary quantum coherence or suppose
that this would take too much time, which could easily be the case if |Ψ〉 isthe state of a more complicated or massive object. Then, what strategy can
Alice and Bob pursue?
As mentioned above, no measurement that Alice can perform on |Ψ〉 willbe sufficient for Bob to reconstruct the state because the state of a quantum
3.2. QUANTUM TELEPORTATION–THE IDEA 35
system cannot be fully determined by measurements. Quantum systems are
so evasive because they can be in a superposition of several states at the same
time. A measurement on the quantum system will force it into only one of
these states; this is often referred to as the projection postulate. We can
illustrate this important quantum feature by taking a single photon, which
can be horizontally or vertically polarized, indicated by the states |H〉 and|V 〉. It can even be polarized in the general superposition of these two states
|Ψ〉 = α |H〉+ β |V 〉 , (3.1)
were α and β are two complex numbers satisfying |α|2 + |β|2 = 1. To place
this example in a more general setting we can replace the states |H〉 and|V 〉 in Eq.(3.1) by |0〉 and |1〉, which refer to the states of any two-state
quantum system. Superpositions of |0〉 and |1〉 are called qubits to signify
the new possibilities introduced by quantum physics into information science
[Schumacher, 1995].
If a photon in state |Ψ〉 passes through a polarizing beamsplitter, a devicethat reflects (transmits) horizontally (vertically) polarized photons, it will be
found in the reflected (transmitted) beam with probability |α|2 (|β|2). Thenthe general state |Ψ〉 has been projected either onto |H〉 or onto |V 〉 bythe action of the measurement. We conclude that the rules of quantum
mechanics, in particular the projection postulate, make it impossible for
Alice to perform a measurement on |Ψ〉 by which she would obtain all the
information necessary to reconstruct the state.
3.2.2 The concept of quantum teleportation
Although the projection postulate in quantum mechanics seems to bring
Alice’s attempts to provide Bob with the state |Ψ〉 to a halt, it was realizedby Bennett et al. [Bennett et al., 1993] that precisely this projection postulate
enables teleportation of |Ψ〉 from Alice to Bob. During teleportation Alice
36 CHAPTER 3. QUANTUM TELEPORTATION
will destroy the quantum state at hand while Bob receives the quantum state,
with neither Alice nor Bob obtaining information about the state |Ψ〉. A key
role in the teleportation scheme is played by an entangled ancillary pair of
particles which will be initially shared by Alice and Bob.
� � �
classicalinformation
ALICE
BOB
EPR-source
teleportedstate
entangled pair
initialstate
BSM
U
Figure 3.1: Scheme showing principle of quantum teleportation. Alice has aquantum system, particle 1, in an initial state which she wants to teleport toBob. Alice and Bob also share an ancillary entangled pair of particles 2 and3 emitted by an Einstein-Podolsky-Rosen(EPR) source. Alice then performsa joint Bell-state measurement (BSM) on the initial particle and one of theancillaries, projecting them also onto an entangled state. After she has sentthe result of her measurement as classical information to Bob, he can performa unitary transformation (U) on the other ancillary particle resulting in itbeing in the state of the original particle.
Suppose particle 1 which Alice wants to teleport is in the initial state
|Ψ〉1 = α |H〉1 + β |V 〉1 (Fig. 3.1), and the entangled pair of particles 2 and
3 shared by Alice and Bob is in the state:
∣∣∣Ψ−⟩23=
1√2(|H〉2 |V 〉3 − |V 〉2 |H〉3) . (3.2)
3.2. QUANTUM TELEPORTATION–THE IDEA 37
That entangled pair is a single quantum system in an equal superposition of
the states |H〉2 |V 〉3 and |V 〉2 |H〉3. The entangled state contains no informa-tion on the individual particles; it only indicates that the two particles will
be in opposite states. The important property of an entangled pair is that as
soon as a measurement on one of the particles projects it, say, onto |H〉 thestate of the other one is determined to be |V 〉, and vice versa. How could
a measurement on one of the particles instantaneously influence the state
of the other particle which, can be arbitrarily far away?! Einstein, among
many other distinguished physicists, could simply not accept this ”Spooky
action at a distance”. But this property of entangled states has now been
demonstrated by numerous experiments (for reviews, see refs. [Clauser and
Shimony, 1978; Greenberger et al., 1993].
The teleportation scheme works as follows. Alice has the particle 1 in the
initial state |Ψ〉1 and the ancillary particle 2. Particle 2 is entangled with theother ancillary particle 3 in the hands of Bob. Although this establishes the
possibility of nonclassical correlations between Alice and Bob, the entangled
pair at this stage contains no information about |Ψ〉1. Indeed the entire
system, comprising Alice’s unknown particle 1 and the entangled pair is in
a pure product state, |Ψ〉1 |Ψ−〉23, involving neither classical correlation nor
quantum entanglement between the unknown particle and the entangled pair.
Therefore no measurement on either member of the entangled pair, or both
together, can yield any information about |Ψ〉1.
The essential point to achieve teleportation is to perform a joint Bell-state
measurement on particles 1 and 2 which projects them onto one of the four
entangled states :
|Ψ±〉12 = 1√2(|H〉1 |V〉2 ± |V〉1 |H〉2)
|Φ±〉12 = 1√2(|H〉1 |H〉2 ± |V〉1 |V〉2) .
(3.3)
Note that these four states are a complete orthonormal basis for particles 1
and 2. The complete state of the three particles before Alice’s measurement
38 CHAPTER 3. QUANTUM TELEPORTATION
is
|Ψ〉123 = α√2(|H〉1 |H〉2 |V〉3 − |H〉1 |V〉2 |H〉3)
+ β√2(|V〉1 |H〉2 |V〉3 − |V〉1 |V〉2 |H〉3) .
(3.4)
In this equation, each direct product |〉1 |〉2 can be expressed in terms of thefour Bell states, and one can thus rewrite Eq.(3.4) as
|Ψ〉123 = 12[|Ψ−〉12 (−α |H〉3 − β |V〉3)
+ |Ψ+〉12 (−α |H〉3 + β |V〉3)+ |Φ−〉12 (α |V〉3 + β |H〉3)+ |Φ+〉12 (α |V〉3 − β |H〉3)] .
(3.5)
It follows that, regardless of the unknown state |Ψ〉1, the four Bell-state
measurement outcomes are equally likely, each occurring with probability
1/4. Quantum physics predicts that once particles 1 and 2 are projected into
one of the four entangled states, particle 3 is instantaneously projected into
one of the four pure states superposed in Eq.(3.5). Denoting |H〉 by the
vector
(10
)and |V 〉 by
(01
), they are thus, respectively,
− |Ψ〉3 ,(−1 00 1
)|Ψ〉3 ,(
0 11 0
)|Ψ〉3 ,
(0 −11 0
)|Ψ〉3 .
(3.6)
where |Ψ〉3 = α |H〉3+β |V 〉3. Each of these possible resultant states for Bob’sEPR particle 3 is related in a simple way to the original state |Ψ〉1 which
Alice sought to teleport. In the case of the first (singlet) outcome, the state of
particle 3 is the same as the initial state of particle 1 except for an irrelevant
phase factor, so Bob need do nothing further to produce a replica of Alice’s
unknown state. In the other three cases, Bob could accordingly apply one of
the unitary transformations in Eq.(3.6) to convert the state of particle 3 into
the original state of particle 1, after receiving via a classical communication
channel the information which of the Bell-state measurement results was
3.2. QUANTUM TELEPORTATION–THE IDEA 39
obtained by Alice. (For the photon polarization state, one can use a suitable
combination of half-wave plates to perform these unitary transformations.)
After Bob’s unitary operation, the final state of particle 3 is therefore
|Ψ〉3 = α |H〉3 + β |V 〉3 . (3.7)
Note that during the Bell-state measurement particle 1 loses its identity
because it becomes entangled with particle 2. Therefore the state |Ψ〉1 is
destroyed on Alice’s side during teleportation.
The result in Eq.(6.2) deserves some further comments. The transfer of
quantum information from particle 1 to particle 3 can happen over arbitrary
distances, hence the name teleportation. Experimentally, quantum entangle-
ment has been shown to survive over distances of the order of 10 km [Tittel
et al., 1998a; 1998b]. We note that in the teleportation scheme it is not
necessary for Alice to know where Bob is. Furthermore, the initial state of
particle 1 can be completely unknown not only to Alice but to anyone. It
could even be quantum mechanically completely undefined at the time the
Bell-state measurement takes place. This is the case when, as already re-
marked by Bennett et al. [Bennett et al., 1993], particle 1 itself is a member
of an entangled pair and therefore has no well-defined properties on its own.
This ultimately leads to entanglement swapping [Zukowski et al., 1993, Bose
et al., 1998].
It is also important to notice that the Bell-state measurement does not
reveal any information on the properties of any of the particles. This is the
very reason why quantum teleportation using coherent two-particle superpo-
sitions works, while any measurement on one-particle superpositions would
fail. The fact that no information whatsoever on either particle is gained
is also the reason why quantum teleportation escapes the verdict of the no-
cloning theorem [Wootters and Zurek, 1982]. After successful teleportation
particle 1 is not available in its original state anymore, and therefore particle
3 is not a clone but really the result of teleportation.
40 CHAPTER 3. QUANTUM TELEPORTATION
3.3 Experimental teleportation
3.3.1 Experimental scheme
Teleportation necessitates both production and measurement of entangled
states; these are the two most challenging tasks for any experimental realiza-
tion. Thus far there are only a few experimental techniques by which one can
prepare entangled states, and there exists no experimentally realized proce-
dure to identify all four Bell-states for any kind of quantum system composed
of two separate particles. However, as discussed in Chapter 2 entangled pairs
of photons can readily be generated and they can be projected onto at least
two of the four Bell states.
Using the technique in section 2.3, we were able to produce the entangled
photons 2 and 3 by type II parametric down-conversion (Fig. 3.2). Inside a
non-linear crystal, an incoming pump photon can decay spontaneously into
two photons which are in the polarization entangled state given by equation
(3.2).
For practical convenience, in the experiment we decided to analyze only
the projection onto |Ψ−〉12 . As discussed in section 2.2.1, this projection is
realized by interfering the two photons, 1 and 2, at a 50 : 50 beam splitter
and, detecting a coincidence between the two detectors at the different out-
put ports of the beam splitter. Here, such a coincident detection acts as a
projection onto |Ψ−〉12. Since originally the polarization state of photon 2 is
completely undetermined, the combined state between photons 1 and 2 is in
an equal superposition of the four Bell states. As a result, in one out of four
cases on average a coincidence will be recorded by the two detectors behind
the beam splitter; that is, a projection onto |Ψ−〉12 takes place.
It is clear that once particles 1 and 2 are projected into |Ψ−〉12, particle
3 is instantaneously projected into |Ψ〉3. Yet we note, with emphasis, that
even we choose to identify only one of the four Bell states, here |Ψ−〉12,
3.3. EXPERIMENTAL TELEPORTATION 41
Figure 3.2: Experimental Setup. A pulse of ultraviolet light passing througha non-linear crystal creates the ancillary pair of entangled photons 2 and 3.After retroflection during its second passage through the crystal the ultravioletpulse creates another pair of photons, one of which will be prepared in theinitial state of photon 1 to be teleported, the other one serves as a triggerindicating that a photon to be teleported is under way. Alice then looks forcoincidences after a beam splitter (BS) where the initial photon and one ofthe ancillaries are superposed. Bob, after receiving the classical informationthat Alice obtained a coincidence count in detectors f1 and f2 identifying the|Ψ−〉12 Bell-state, knows that his photon 3 is in the initial state of photon 1which he then can check using polarization analysis with the polarizing beamsplitter (PBS) and the detectors d1 and d2. The detector P provides theinformation that photon 1 is under way.
42 CHAPTER 3. QUANTUM TELEPORTATION
teleportation is successfully achieved, albeit only in a quarter of the cases.
As mentioned already, if we further insert a two-channel polarizer into each of
the outputs of the beam splitter BS, then a coincident detection between the
two outputs of the polarizer on either side of the BS acts as a projection onto
|Ψ+〉12. Thus, a slight change of our scheme is actually capable of achieving
teleportation in 50% of the time–in those occasions when Alice happened to
detect state |Ψ−〉12 or |Ψ+〉12.
Note that in our teleportation scheme the Bell-state analysis relies on
the interference of two independently created photons. One, therefore, has
to guarantee good spatial and temporal overlap at the beam splitter and,
above all, one has to erase all kinds of path information for photons 1 and 2.
Especially the strong time and frequency correlations of the two photons 2
and 3 created by parametric down-conversion can give rise to Welcher-Weg
information for the interfering photons [Herzog et al., 1995].
There are two possibilities for quantum erasure. In the first one, Welcher-
Weg information is erased by detecting photons 1 and 2 within time intervals
much shorter than their coherence time [Zukowski et al., 1993]. Then, such
ultracoincident registrations are too close in time to discriminate which of
the detected photons shares the source with photon 3 or with photon 4,
respectively. Yet, this method cannot be used in practice due to the poor
time resolution of the existing single-photon detectors (typically 0.5ns for
Si-avalanche photodiodes as compared to typical coherence times of about
500fs).
The second possibility involves increasing the coherence times of the in-
terfering photons to become much longer than the time interval within which
they are created [Zukowski et al., 1995]. Then again, one cannot infer any-
more which of the detected photons 1 or 2 was created together with photon
3, or with photon 4, respectively. In our experiment UV pulses with a du-
ration of 200fs are used to create the photon pairs. We then chose narrow
bandwidth filters (∆λ = 4.6nm) in front of the detectors registering photons
1 and 2. The resulting coherence time of about 500fs is sufficiently longer
3.3. EXPERIMENTAL TELEPORTATION 43
than the pump pulse duration. Furthermore, single-mode fiber couplers act-
ing as spatial filters were used to guarantee good mode overlap of the detected
photons.
Figure 3.2 is a schematic drawing of the experimental setup. UV pulses
are produced by frequency doubling the pulses of a commercial mode locked
Ti:sapphire laser from 788 to 394nm using a nonlinear LBO crystal (LiB3O5).
For a repetition rate of 76 MHz we obtained an averaged power of 500 mW.
Passing the UV pulses through a BBO crystal (β−BaB2O4) creates via type-
II down-conversion a pair of photons, 2 and 3, in the entangled state |Ψ−〉23.
After reflection, the pump pulse passes the crystal again and produces the
second pair of photons 1 and 4. Photon 1 is prepared in the initial state to be
teleported, and its partner, photon 4, serves to indicate that it was emitted.
How can one experimentally prove that an arbitrary unknown quantum
state can be teleported? First, one has to show that teleportation works for
a (complete) basis, a set of known states into which any other state can be
decomposed. A basis for polarization states has just two components, and
in principle we could choose as the basis horizontal and vertical polarization
as emitted by the source. Yet this would not demonstrate that teleporta-
tion works for any general superposition, because these two directions are
preferred directions in our experiment. Therefore, in the first demonstration
we chose as the basis for teleportation the two states linearly polarised at
-45◦ and +45◦ which are already superpositions of the horizontal and vertical
polarizations. Second, one has to show that teleportation works for superpo-
sitions of these base states. Therefore we also demonstrate teleportation for
circular polarisation. This covers all three mutually orthogonal axes of the
Poincare sphere.
3.3.2 Results
In the first experiment photon 1 is polarized at 45◦. Teleportation should
work as soon as photon 1 and 2 are detected in the |Ψ−〉12 state, which occurs
44 CHAPTER 3. QUANTUM TELEPORTATION
in 25% of all possible cases. The |Ψ−〉12 state is identified by recording a
coincidence between two detectors, f1 and f2, placed behind the beamsplitter
(Fig. 3.2).
If we detect a f1f2 coincidence (between detectors f1 and f2), then photon
3 should also be polarized at 45◦. The polarization of photon 3 is analysed by
passing it through a polarizing beamsplitter selecting +45◦ and −45◦ polar-ization. To demonstrate teleportation, only detector d2 at the +45◦ output
of the polarizing beamsplitter should click (that is, register a detection) once
detectors f1 and f2 click. Detector d1 at the −45◦ output of the polarizingbeamsplitter should not detect a photon. Therefore, recording a three-fold
coincidence d2f1f2 (+45◦ analysis) together with the absence of a three-fold
coincidence d1f1f2 (−45◦ analysis) is a proof that the polarization of photon1 has been teleported to photon 3.
To meet the condition of temporal overlap, we change in small steps the
arrival time of photon 2 by changing the delay between the first and second
down conversion by translating the retroflection mirror (Fig. 3.2). In this
way we scan into the region of temporal overlap at the beamsplitter so that
teleportation should occur.
Outside the region of teleportation photon 1 and 2 each will go either to f1
or to f2 independent of one another. The probability to have a coincidence
between f1 and f2 is therefore 50%, which is twice as high as inside the
region of teleportation. Photon 3 should not have a well-defined polarization
because it is part of an entangled pair. Therefore, d1 and d2 have both
a 50% chance of receiving photon 3. This simple argument yields a 25%
probability both for the −45◦ analysis (d1f1f2 coincidences) and for the +45◦
analysis (d2f1f2 coincidences) outside the region of teleportation. Figure 3
summarizes the predictions as function of the delay. Successful teleportation
of the +45◦ polarization state is then characterized by a decrease to zero in
the −45◦ analysis (Fig. 3.3a), and by a constant value for the +45◦ analysis(Fig. 3.3b).
3.3. EXPERIMENTAL TELEPORTATION 45
-100 -50 0 50 100
(b)+45°
Theory: +45° Teleportation3
-fo
ldc
oin
cid
en
ce
pro
ba
bili
ty
Delay (µm)
0,00
0,05
0,10
0,15
0,20
0,25
0,00
0,05
0,10
0,15
0,20
0,25
(a)-45°
Figure 3.3: Theoretical prediction for the three-fold coincidence probabilitybetween the two Bell-state detectors (f1, f2) and one of the detectors analysingthe teleported state. The signature of teleportation of a photon polarizationstate at +45◦ is a dip to zero at zero delay in the three-fold coincidence ratewith the detector analysing -45◦ (d1f1f2) (a) and a constant value for thedetector analysis +45◦ (d2f1f2) (b). The shaded area indicates the region ofteleportation.
46 CHAPTER 3. QUANTUM TELEPORTATION
The theoretical prediction of Fig. 3.3 may easily be understood by real-
izing that at zero delay there is a decrease to half in the coincidence rate
for the two detectors of the Bell-state analyser, f1 and f2, as compared to
outside the region of teleportation. Therefore, if the polarization of photon 3
were completely uncorrelated to the others the three-fold coincidence should
also show this dip to half. That the right state is teleported is indicated by
the fact that the dip goes to zero in Fig. 3.3a and it is filled to a flat curve
in Fig. 3.3b.
We note that about as likely as production of photons 1, 2, and 3 is emis-
sion of two pairs of down-converted photons by a single source. Although
there is no photon coming from the second source (photon 1 is absent), there
will still be a significant contribution to the three-fold coincidence rates.
These coincidences have nothing to do with teleportation and can be identi-
fied by blocking the path of photon 1.
The probability for this process to yield spurious two- and three-fold co-
incidences can be estimated by taking into account the experimental param-
eters. The experimentally determined value for the percentage of spurious
three-fold coincidences is 68%± 1%. In the experimental graphs of Fig. 3.4
we have subtracted the experimentally determined spurious coincidences.
The experimental results for teleportation of photons polarized under
+45◦ is shown in the left column of Fig. 3.4; Fig. 3.4a and b should be
compared with the theoretical predictions as shown in Fig. 3.3. The strong
decrease in the −45◦ analysis, and the constant signal for the +45◦ analysis,indicate that photon 3 is polarized along the direction of photon 1, confirming
teleportation.
The results for photon 1 polarized at −45◦ demonstrate that teleporta-tion works for a complete basis for polarization states (right-hand column
of Fig. 3.4). To rule out any classical explanation for the experimental re-
sults, we have produced further confirmation that our procedure works by
additional experiments. In these experiments we teleported photons linearly
3.3. EXPERIMENTAL TELEPORTATION 473
-fo
ldc
oin
cid
en
ce
sp
er
20
00
sec
on
ds
(c)
-45° Teleportation+45° Teleportation
0
100
200
300
400
500
600
0
100
200
300
400
500
(a)
(b)
-150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150
0
100
200
300
400
500
600
0
100
200
300
400
500
(d)
Delay (µm) Delay (µm)
-45°
+45°
-45°
+45°
Figure 3.4: Experimental results. Measured three-fold coincidence ratesd1f1f2 (-45◦) and d2f1f2 (+45◦) in the case that the photon state to be tele-ported is polarized at +45◦ (a and b) or at -45◦ (c and d). The coincidencerates are plotted as function of the delay (in µm) between the arrival of pho-ton 1 and 2 at Alice’s beamsplitter (see Fig.1.2). The three-fold coincidencerates are plotted after subtracting the spurious three-fold background contri-bution (see text). These data, compared with Fig.1.3, together with similarones for other polarisations (Table 1) confirm teleportation for an arbitrarystate.
48 CHAPTER 3. QUANTUM TELEPORTATION
polarized at 0◦ and at 90◦, and also teleported circularly polarized photons.
The experimental results are summarized in Table 1, where we list the visibil-
ity of the dip in three-fold coincidences, which occurs for analysis orthogonal
to the input polarization.
polarization visibility+45◦ 0.63± 0.02−45◦ 0.64± 0.020◦ 0.66± 0.0290◦ 0.61± 0.02
circular 0.57± 0.02
Table 1
As mentioned above, the values for the visibilities are obtained after sub-
tracting the offset caused by spurious three-fold coincidences. These can
experimentally be excluded by conditioning the three-fold coincidences on
the detection of photon 4, which effectively projects photon 1 into a single-
particle state. We have performed this four-fold coincidence measurement for
the case of teleportation of the +45◦ and +90◦ polarization states, that is, for
two non-orthogonal states. The experimental results are shown in Fig. 3.5.
Visibilities of 70%± 3% are obtained for the dips in the orthogonal polariza-
tion states! Here, these visibilities are directly the degree of polarization of
the teleported photon in the right state without any background subtracted.
As can be seen from the measured visibilities, the teleportation fidelity is
rather high in our experiment. Typically, it is of the order of 0.85. This very
clearly surpasses the limit of 2/3 [Massar and Popescu, 1995] which at best
could have been obtained by Alice performing a polarisation measurement on
the given photon, informing Bob about the measurement result via classical
communication, and by Bob repreparing the photon state at his output.
The measured high fidelity proves that we demonstrated teleportation of the
quantum state of a single photon.
3.3. EXPERIMENTAL TELEPORTATION 49
(c)(a)
(b) (d)
-45° 0°
+45° +90°
-150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150
4-f
old
co
inc
ide
nc
es
pe
r4
00
0se
co
nd
s
90° Teleportation45° Teleportation
Delay (µm) Delay (µm)
0
20
40
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
120
0
20
40
60
80
100
120
Figure 3.5: Four-fold coincidence rates (without background subtraction).Conditioning the three-fold coincidences as shown in Fig.1.4 on the registra-tion of photon 4 (see Fig.1.2) eliminates the spurious three-fold background.Graphs a and b show the four-fold coincidence measurements for the caseof teleportation of the +45◦ polarization state, and graphs c and d show theresults for the +90◦ polarization state. The visibilities, and thus the polariza-tions of the teleported photons, obtained without any background subtractionare 70% ± 3%. These results for teleportation of two non-orthogonal statesprove that we demonstrated teleportation of the quantum state of a singlephoton.
50 CHAPTER 3. QUANTUM TELEPORTATION
3.4 Discussion
In our experiment, we used pairs of polarization entangled photons as pro-
duced by pulsed down-conversion and two-photon interferometric methods
to transfer the polarization state of one photon onto another one. But, tele-
portation is by no means restricted to this system. In addition to pairs of
entangled photons or entangled atoms [Hagley 1997, Fry 1995], one could
imagine entangling photons with atoms or photons with ions, and so on.
Then teleportation would allow us to transfer the state of, for example, fast-
decohering, short-lived particles onto some more stable systems. This opens
the possibility of quantum memories, where the information of incoming pho-
tons is stored on trapped ions, carefully shielded from the environment.
Furthermore, entanglement purification [Bennett et al., 1996a] is a scheme
of improving the quality of entanglement if it was degraded by decoherence
during storage or transmission of the particles over noisy channels. Then it
becomes possible to send the quantum state of a particle to some place, even if
the available quantum channels are of very poor quality and thus sending the
particle itself would very probably destroy the fragile quantum state. The
feasibility of preserving quantum states in a hostile environment will have
great advantages in the realm of quantum computation. The teleportation
scheme can be used to provide links between quantum computers.
Quantum teleportation is not only an important ingredient in quantum
information tasks; it also allows new types of experiments and investigations
on the foundations of quantum mechanics. As any arbitrary state can be
teleported, so can the fully undetermined state of a particle which is member
of an entangled pair. By doing so, one can transfer the entanglement between
particles. This allows us not only to chain the transmission of quantum
states over distances, where decoherence would have already destroyed the
state completely, but it also enables us to perform a test of Bell’s theorem on
particles which do not share any common past, a new step in the investigation
of the features of quantum mechanics. Last but not least, novel experiments
3.4. DISCUSSION 51
disproving concepts of a the local realistic character of nature become possible
if one uses features of the experiment presented here to generate entanglement
between more than two spatially separated particles [Greenberger et al., 1990;
Zeilinger et al., 1997]. A first such experiment will be reported in Chapter 6.
Since our first experimental realization, quantum teleportation has been
one of the main focus in the field of quantum information physics. In the past
two years, several groups in the world have achieved a number of experimental
advances in quantum teleportation. It is worth comparing these experiments
with our scheme and, briefly discussing the differences among them.
Though Popescu’s suggestion, as realized by De Martini’s group in Rome
[Boschi et al., 1998], provides an elegant way to demonstrate some relevant
Hilbert-space formalism in the absence of a two-photon Bell measurement,
it is not quite proper to claim that the Rome experiment does constitute
teleportation. What really constitutes teleportation? An essential criterion
is to be able to teleport any independent quantum state coming from outside.
This is obviously not possible in the Rome experiment, where the initial
photon has to be entangled from the beginning with the final one. One might
argue that the scheme by Popescu is equivalent to the original teleportation
scheme up to a local operation since, in principle, any unknown state of
a particle from outside could be swapped onto the polarization degree of
freedom of Alice’s EPR particle by a local unitary operation. However, such
a local unitary operation would require a quantum C-NOT gate that does not
exist yet. Further we note, with emphasis, that an experimental realization
of quantum C-NOT gate itself leads a complete Bell measurement and thus
a full realization of the original teleportation scheme, Popescu’s scheme is
therefore not necessary anymore.
Generally speaking, the basic criterions to constitute a bona fide telepor-
tation should be (1) the experimental scheme is capable of teleporting any
state that is designed to teleport, (2) a fidelity better than 2/3 [Massar and
Popescu, 1995], (3) for future applications, at least in principle, the scheme
should be able to be extended to long-distance teleportation, that is, the real
52 CHAPTER 3. QUANTUM TELEPORTATION
teleportation. Indeed, at the moment only the interference of independently
created photons makes teleportation of any independent and even undefined
(not just simply unknown) photon state possible. Our realization of entan-
glement swapping reported in Chapter 4 will underline the quantum nature
of our teleportation procedures.
The two most recent teleportation experiments were reported by Kim-
ble’s group [Furusawa et al., 1998] and Laflamme’s group [Nielson et al.,
1998], respectively. An obvious advantage of these two schemes is that the
input quantum state can be, in principle, teleported with an efficiency close
to 100%. However, it should be pointed out that while a rather low fidelity
(0.58) was observed in the experiment of Furusawa et al., their scheme is
hardly to be extended to long-distance case because of the unavoidable dis-
persion of squeezed-states during the distribution of entanglement, which
consequently leads to a fast degrading of the quality of squeezed-state entan-
glement. Finally, it is also worthwhile noting that the NMR method used by
Nielson et al. can never teleport a quantum state over macroscopic distance.
Chapter 4
Entanglement Swapping
4.1 Introduction
The phenomenon of entanglement is a remarkable feature of quantum the-
ory. It plays a crucial role in the discussions of the Einstein-Podolsky-Rosen
paradox, of Bell’s inequalities, and of the non-locality of quantum mechanics.
Thus far entanglement has either been realized by having the two entangled
particles emerge from a common source [Freedman 1972, Rarity 1990], or
from having two particles interact either directly or indirectly with each other
[Lamehi-Rachti and Mittig, 1976; Hagley et al., 1997]. Yet, an alternative
possibility to obtain entanglement is to make use of a projection of the state
of two particles onto an entangled state. This projection measurement does
not necessarily require a direct interaction between the two particles: When
each of the particles is entangled with one other partner particle, an appro-
priate measurement, for example, a Bell-state measurement, of the partner
particles will automatically collapse the state of the remaining two particles
into an entangled state. This striking application of the projection postu-
late is referred to as entanglement swapping [Zukowski et al., 1993]. Here,
we report the first experimental realization for entanglement swapping [Pan
1998b]. In our experiment we take two pairs of polarization entangled pho-
tons and subject one photon from each pair to a Bell-state measurement.
53
54 CHAPTER 4. ENTANGLEMENT SWAPPING
This results in projecting the other two outgoing photons into an entangled
state.
4.2 Theoretical scheme
Consider the arrangement of Fig. 4.1. There are two EPR sources, each one
simultaneously emitting a pair of entangled particles. In anticipation to our
experiments we assume that these are polarization entangled photons in the
state
|Ψ〉1234 = 12(|H〉1 |V〉2 − |V〉1 |H〉2)× (|H〉3 |V〉4 − |V〉3 |H〉4)
. (4.1)
Here |H〉 and |V〉 indicate the states of a horizontally and a vertically polar-ized photon, respectively. The total state describes the fact that photons 1
and 2 are entangled in a singlet state in polarization and photons 3 and 4 are
also entangled in the singlet state. Yet, the state of pair 1-2 is factorizable
from the state of pair 3-4, that is, there is no entanglement of any of the
photons 1 or 2 with any of the photons 3 or 4.
We now perform a joint Bell-state measurement on photons 2 and 3, that
is, photons 2 and 3 are projected into one of the four Bell states which form
a complete basis for the combined state of photons 2 and 3
|Ψ±〉23 = 1√2(|H〉2 |V〉3 ± |V〉2 |H〉3)
|Φ±〉23 = 1√2(|H〉2 |H〉3 ± |V〉2 |V〉3) .
(4.2)
This measurement projects photons 1 and 4 also onto a Bell state, a different
one depending on the result of the Bell-state measurement for photons 2 and
3. To consider a specific example let us assume that the result of the Bell-
state measurement of photons 2 and 3 is |Ψ−〉 then it can be seen that the
resulting state for photons 1 and 4 is also |Ψ−〉. In fact, close inspection showsthat for the initial state given in Eq. (4.1) the emerging state of photons 1
4.2. THEORETICAL SCHEME 55
Figure 4.1: Principle of entanglement swapping. Two EPR sources producetwo pairs of entangled photons, pair 1-2 and pair 3-4. One photon fromeach pair (photon 2 and 3) is subjected to a Bell-state measurement (BSM).This results in projecting the other two outgoing photons 1 and 4 into anentangled state. Change of the shading of the lines indicates a change in theset of possible predictions that can be made.
and 4 will be identical to the one photon 2 and 3 collapsed into. This is a
consequence of the fact that the state of Eq. (4.1) can be rewritten as
|Ψ〉1234 = 12(|Ψ+〉14 |Ψ+〉23 − |Ψ−〉14 |Ψ−〉23
− |Φ+〉14 |Φ+〉23 + |Φ−〉14 |Φ−〉23). (4.3)
In all cases photons 1 and 4 emerge entangled despite the fact that they
never interacted in the past. In Fig. 4.1 entangled particles are indicated by
the same line darkness. Note that particles 1 and 4 become entangled after
the Bell-state measurement (BSM) on particles 2 and 3.
The result above can also be interpreted as teleportation of the unknown
state of, say, photon 2 onto photon 4 [Bennett et al., 1993]. In that case
one could consider Alice performing the Bell-state measurement on photons
2 and 3, telling Bob, who is in possession of photon 4, the result of the
Bell-state measurement. Then, by performing one of a fixed set of unitary
operations on photon 4, photons 1 and 4 will be left in an singlet state,
56 CHAPTER 4. ENTANGLEMENT SWAPPING
which is exactly the same as the state of photons 1 and 2 before the Bell-
state measurement. It is conceptually most interesting to realize that in this
case the teleported photon state does not have any well-defined polarization,
because it is entangled with photon 1. It is fair to say that here we do not
teleport some unknown state of a photon but rather an in principle undefined
state. The state of photon 2, and therefore also of the teleported photon 4,
is certainly undefined before any measurements are performed on photons 1
or 4.
It is worthwhile noting that the process of entanglement swapping also
gives a means to define that an entangled pair of photons, 1 and 4, is avail-
able. As soon as Alice completes the Bell-state measurement on particles 2
and 3, we know that photons 1 and 4 are on their way ready for detection in
an entangled state. An experimental realization of entanglement swapping
will thus give, for the first time, the possibility to perform a test of Bell’s
inequality using a pair of photons that never interacted. That is a big step
towards the final realization of so called ”event-ready detections” of the en-
tangled particles, a concept suggested by John Bell [Clauser and Shimony,
1978; Bell, 1980].
4.3 Experimental entanglement swapping
As in our teleportation experiment, here we also analyzed only the projection
onto |Ψ−〉23. Figure 4.2 is a schematic drawing of the experimental setup. A
UV laser pulse passing through a BBO crystal creates via type-II parametric
down-conversion the first pair of entangled photons, 1 and 2, in the state
|Ψ−〉12. After reflection the pump pulse passes the crystal again and creates
the second pair of photons, 3 and 4, in the state |Ψ−〉34. Note that the two
pairs are created independently of one another although the same pulse and
the same crystal are used twice. Photon 2 and 3 are subjected to a Bell-
state measurement (BSM). In order to project the state of photons 2 and 3
onto the anti-symmetric Bell-state |Ψ−〉23, the same technique developed for
4.3. EXPERIMENTAL ENTANGLEMENT SWAPPING 57
quantum teleportation has been used in the experiment.
According to the entanglement swapping scheme, upon projection of pho-
ton 2 and 3 into the |Ψ−〉23 state, photon 1 and 4 should be projected into
the |Ψ−〉14 state. To verify that this entangled state is obtained we have to
analyze the polarization correlation between photons 1 and 4 conditioned on
coincidences between the detectors of the Bell-state analyzer. If photon 1
and 4 are in the |Ψ−〉14 state their polarizations should be orthogonal upon
detection in any polarization basis. Using a λ/2 retardation plate at 22.5◦
and two detectors (D1+ and D1−) behind a polarizing beamsplitter we chose
to analyze the polarization of photon 1 both along the +45◦ axis (D1+) and
along the −45◦ axis (D1−). Photon 4 is analyzed by detector D4 at the
variable polarization direction Θ.
If entanglement swapping happens, then both the two-fold coincidences
between D1+ and D4, and between D1− and D4, conditioned on the |Ψ−〉23
detection, should show two sine curves as a function of Θ which are 90◦
out of phase. The D1+D4 curve should, in principle, go to zero for Θ =
45◦ whereas the D1−D4 curve should show a maximum at this position.
Figure 4.3 shows the experimental results for the coincidences between D1+
and D4, and between D1− and D4, given that photons 2 and 3 have been
registered by the two detectors in the Bell-state analyzer. Note that this
method requires four-fold coincidences. The result clearly demonstrates the
expected sine curves, complementary for the two detectors (D1+ and D1−)
of photon 1 registering orthogonal polarizations. We verified by additional
measurements that the sine curves are independent (up to the corresponding
shift in Θ) on the detection basis of photon 1, that is, independent of the
rotation angle of the λ/2 retardation plate. In other words, the observed
sinusoidal behavior of the coincidence rates depends only on the relative
angle between the polarizers in beams 1 and 4.
The experimentally obtained four-fold coincidences have been fitted by a
joint sine function with the same amplitudes for both curves. Note that the
observed visibility of 0.65 clearly surpasses the 0.5 limit of a classical wave
58 CHAPTER 4. ENTANGLEMENT SWAPPING
Figure 4.2: Experimental setup. A UV-pulse passing through a non-linearcrystal creates pair 1-2 of entangled photons. Photon 2 is directed to thebeamsplitter (BS). After reflection, during its second passage through thecrystal the UV-pulse creates a second pair 3-4 of entangled photons. Photon3 will also be directed to the beamsplitter to perform a Bell-state measure-ment (BSM) of photons 2 and 3. When photons 2 and 3 yield a coincidenceclick at the two detectors behind the beamsplitter a projecting into the |Ψ−〉23
state takes place. As a consequence of this Bell-state measurement the tworemaining photons 1 and 4 will also be projected onto an entangled state. Toanalyse their entanglement we look at coincidences between detectors D1+
and D4, and between detectors D1− and D4, for different polarization anglesΘ. By rotating the λ/2 plate in front of the polarizing beamsplitter (PBS) wecan also analyze photon 1 in a different orthogonal polarization basis which isnecessary to obtain statements for relative polarization angles between pho-tons 1 and 4. Note that, since the detection of coincidences between detectorsD1+ and D4, and D1− and D4 are conditioned on the detection of the Ψ−
state, we are looking at 4-fold coincidences. Narrow bandwidth filters (F) arepositioned in front of each detector.
4.3. EXPERIMENTAL ENTANGLEMENT SWAPPING 59
theory. A visibility of 0.72±0.04 was observed in a few initial measurements
for analysis along 45o. A future experiment for showing a significant violation
of Bell’s inequalities requires a stable visibility better than 0.71.
In order to find a practical way to improve the visibility of four-photon
interference fringes, let us perform a semi-quantitative estimation of the vis-
ibility via our experimental parameters. If we simply assume that our EPR
pairs are of perfect entanglement quality, i.e. a visibility of 100% for two
entangled photons, then according to Zukowski et al. [Zukowski et al., 1995;
Zeilinger et al., 1997] the visibility of four-photon fringes is given by
Videal =σP√
σ2P + σ2
F
(4.4)
where σP and σF are the spectral width of the pump pulse and the bandwidth
of the interference filters, respectively. Our narrow bandwidth filters σF ≈4.6nm and the measured pulse spectral width σP ≈ 8nm yield Videal ≈ 87%.
However, as observed in the experiment the EPR pairs produced by pulsed
pump have at best a visibility of VEPR ≈ 90% and cannot fully satisfy the re-
quirement of perfect entanglement. Therefore, one must also consider such a
degrading effect on the experimental visibility of four-photon fringes. Because
we observe interference fringes between two independent pairs, one possible
way to estimate the visibility may be the product Videal× V 2EPR. This conse-
quently leads to an estimated visibility of 87%× 90% × 90% ≈ 72%, which
compares favourably with our best experimental results.
Our estimation indicates two possible ways to improve the visibility be-
yond the 0.71 limit. In principle, one could use interference filters with more
narrow bandwidth or try to produce EPR pairs with higher quality of entan-
glement, though both approaches are technically certainly challenging due to
the very low coincidence rate. Therefore, we also expect to improve the very
low four-fold coincidence rate, the main difficulty of the present experiment,
by using a new laser system currently being installed in Vienna, leading to
60 CHAPTER 4. ENTANGLEMENT SWAPPING
a better performance of the experiment.
Figure 4.3: Verification of entanglement swapping via verification of entan-glement between the two photons 1 and 4 from separate pairs. Four-fold co-incidences, resulting from two-fold coincidence D1+D4 (D1−D4) conditionedon the Bell-state measurement two-fold coincidences at detectors D2 andD3, as a function of the polarization angle Θ. The two complementary sinecurves with a visibility of 0.65 ± 0.02 demonstrate that photons 1 and 4 arepolarization entangled.
Again, we mention that, obviously, registration of a coincidence in the
two detectors behind the beam splitter could also have been caused by two
pairs created in either source. That possibility could clearly be ruled out by
sophisticated detection procedures. It certainly does not have any implication
on those events in our experiment where we indeed obtain four registration
events.
4.4. GENERALIZATION AND APPLICATIONS 61
4.4 Generalization and applications
While experimental entanglement swapping itself is a further demonstration
of teleportation, i.e. teleportation of quantum mechanically undefined state,
a test of Bell’s theorem using entanglement swapping could test nonlocality
with a pair of particles that never interacted. Such a test would certainly help
us to further understand nonlocality. We could even perform a delayed choice
experiment for entanglement swapping as recently suggested by Peres [Peres,
1999], where one could delay the instant of time of the to perform Bell-state
measurement on photons 2 and 3, and thus entanglement between photons
1 and 4 is produced a posteriori, after they have already been measured and
may no longer exist.
Further, various generalizations of the present scheme are at hand [Bose
1997, Pan 1998a, 1999b]. One could have many different kinds of entan-
glement to begin with, perform various different measurements, and obtain
novel kinds of entanglement for the emerging particles. A first clear possi-
bility [Zukowski et al., 1995; Rarity and Tapster, 1995] is to project three
particles, each from an entangled pair, into a GHZ state [Greenberger et al.,
1990] whenceforth the other three emerging particles are also projected into
a GHZ state.
Secondly, for example, one could use a polarizing beam splitter instead
of the beam splitter in the set-up (Fig. 4.2). Then, using the same reasoning
line as used in section 2.2.2 (or more directly refer to Fig. 2.7 and Eq.(2.33))
we could arrive the following conclusions:
(1) The outgoing four-particle state will be in a conditional GHZ state
immediately after photons 2 and 3 passing through the polarizing beam split-
ter;
(2) as suggested in chapter 2, one could perform ±45◦ polarization anal-ysis at one of the two output ports of the polarizing beam splitter, then
conditioned on the detection of a single photon with 45◦ (or −45◦) polariza-tion, the remaining three photons will be correspondingly projected into a
62 CHAPTER 4. ENTANGLEMENT SWAPPING
freely propagating GHZ-state;
(3) one could also perform ±45◦ polarization analysis at the two outputports in the meantime, that is, perform a Bell-state measurement on photons
2 and 3 by using our modified Bell-state analyzer (refer to Fig. 2.5), then by
Eq. (4.3) photons 1 and 4 will be projected into state |Φ+〉14 (or |Φ−〉14) with
respect to a projection of the state of photons 2 and 3 into the Bell-state
|Φ+〉23 (or |Φ−〉23).
The results above exactly imply that one could interpret our experimental
setup in different ways, corresponding to what kind of specific measurement
one intends to perform. It should be noted that such a scheme can be easily
generalized to N particles case by just adding more EPR pairs and polarizing
beam splitters. Again all these schemes above require pulsed pump technol-
ogy, with pulses of even higher power than in the present experiment, yet in
principle achievable with current technology.
We might also remark that the present results, taken together with our
verification of quantum teleportation in chapter 3, are easily understood
in the framework of the Copenhagen interpretation of quantum mechanics
[Nagel, 1989]. They cause no conceptual problems if one accepts that in-
formation about quantum systems is a more basic feature than any possible
”real” properties these systems might have [Zeilinger, 1998].
Finally, it is foreseen that entanglement swapping, besides its interest to
fundamental physics, will have a number of important applications in future
quantum communication schemes. First of all, as mentioned by Bose et al.
[Bose et al., 1998], our entanglement swapping scheme opens up a way to
speed up the distribution of entanglement for any particles possessing mass.
The importance of the distribution of entanglement between distant parties
is obvious as Bell pairs are essential for the implementation of many quantum
communication schemes over large distance, such as secret-key distribution
[Ekert, 1991], teleportation [Bennett et al., 1993] and dense coding [Bennett
and Wiesner, 1992].
4.4. GENERALIZATION AND APPLICATIONS 63
On the other hand, due to the unavoidable decoherence caused by cou-
pling with the environment, the quality of entanglement will be degraded dur-
ing the distribution and storage of entanglement. Therefore, entanglement
purification [Bennett et al., 1996a; 1996b; Deutsch et al., 1996] is of great sig-
nificance to achieve quantum communication with perfect fidelity. Because
all purification schemes involve collective measurements on many photons at
once, or need two-qubit logic gates for polarization-entangled photons, whose
physical implementation is very difficult under the current technology, and
remains to be realized in an experiment. Furthermore, for distances much
larger than the coherence length of a corresponding noisy quantum channel,
the fidelity of transmission is so low that the standard purification methods
above are not applicable. However, while the most recent research [Bose et
al., 1999] shows that a simple variant of our entanglement swapping scheme
can be directly used to purify single pairs of polarization-entangled photons,
it is possible to divide the quantum channel into shorter segments that are
purified separately and then connected by entanglement swapping [Briegel
et al, 1998; Duer et al., 1999]. All this underlines that entanglement swap-
ping is one of the most important key procedures in quantum communication
networks.
Chapter 5
Three-photon GHZentanglement
5.1 Introduction
Ever since the seminal work of Einstein, Podolsky and Rosen [Einstein 1935]
there has been a quest for generating entanglement between quantum par-
ticles. Although two-particle entanglements have long been demonstrated
experimentally [Wu and Shaknov, 1950; Freedman and Clauser, 1972; As-
pect et al., 1982; Kwiat et al., 1995; Hagley et al., 1997], the preparation
of entanglement between three or more particles remains an experimental
challenge. Proposals have been made for experiments with photons [Green-
berger et al., 1990; Zeilinger et al., 1997; Pan and Zeilinger, 1998a] and atoms
[Cirac and Zoller, 1994; Haroche, 1995], and three nuclear spins within a sin-
gle molecule have been prepared such that they locally exhibit three-particle
correlations [Lloyd, 1998; Laflamme et al., 1998]. However, until now there
has been no experiment which demonstrates the existence of entanglement of
more than two spatially separated particles. Here we present the first experi-
mental observation of polarization entanglement of three spatially separated
photons [Bouwmeester, Pan et al., 1999]. Such states, known as Greenberger-
Horne-Zeilinger (GHZ) states, are interesting from both a fundamental and
64
5.2. EXPERIMENTAL SET-UP 65
a technological point of view.
The original motivation to prepare three-particle entanglements stems
from the observation by Greenberger, Horne and Zeilinger that three-particle
entanglement leads to a conflict with local realism for non-statistical predic-
tions of quantum mechanics [Greenberger et al., 1989; 1990; Mermin, 1990a;
1990b]. This is in contrast to the case of Einstein-Podolsky-Rosen experi-
ments with two entangled particles testing Bells inequalities, where the con-
flict only arises for the statistical predictions of quantum theory [Bell, 1964].
The incentive to produce GHZ states has been significantly increased by
the advance of the field of quantum communication and quantum information
processing. Entanglement between several particles is the most important
feature of many such quantum communication and computation protocols
[Bennett, 1995].
5.2 Experimental Set-up
The experiment described here is based on techniques that have been devel-
oped for our previous experiments on quantum teleportation [Bouwmeester,
Pan et al., 1997] and entanglement swapping [Pan et al., 1998]. In fact, one
of the main complications in the those experiments, namely, the creation of
two pairs of photons by a single source, is here turned into a virtue.
The main idea, as was put forward in [Zeilinger et al., 1997; Pan and
Zeilinger, 1998], is to transform two pairs of polarization entangled photons
into a triplet of entangled photons and a fourth independent photon. Our
experimental arrangement is such that we start with two pairs of entangled
photons and register the photons in a way that any information as to which
pair each photon belongs to is erased. Fig. 5.1 is a schematic drawing of our
experimental setup. Pairs of polarization-entangled photons are generated
by a short pulse of ultraviolet (UV) light (≈ 200 fs, λ = 788 nm from a
frequency-doubled, mode-locked Ti-Sapphire laser), which passes through an
66 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT
optically nonlinear crystal (Beta-Barium-Borate, BBO). The probability per
pulse to create a single pair in the desired modes is rather low and of the
order of a few 10−4. The pair creation is such that the following polarization
entangled state is obtained [Kwiat et al., 1995]:
1√2(|H〉a | V 〉b − |V 〉a |H〉b) . (5.1)
This state indicates that there is a superposition of the possibility that the
photon in arm a is horizontally polarized and the one in arm b vertically
polarized (|H〉a |V 〉b), and the opposite possibility, i.e., |V 〉a |H〉b. The mi-nus sign indicates that there is a fixed phase difference of π between the two
possibilities. For our GHZ experiment this phase factor is actually allowed
to have any value, as long as it is the same for both pairs.
The setup is such that arm a continues towards a polarizing beam split-
ter, where V photons are reflected and H photons are transmitted towards
detector T (behind an interference filter δλ = 4.6 nm at 788 nm). Arm b con-
tinues towards a 50/50 polarization-independent beam splitter. From each
beam splitter, one output is directed to a final polarizing beam splitter. In
between the two polarizing beam splitters there is a λ/2 wave plate at an an-
gle of 22.5◦ which rotates the vertical polarization of the photons reflected by
the first polarizing beam splitter into a 45◦ polarization, i.e. a superposition
of |H〉 and |V 〉 with equal amplitudes. The remaining three output arms
continue through interference filters (δλ = 3.6 nm) and single-mode fibers
towards the single-photon detectors D1, D2, and D3. Including filter losses,
coupling into single-mode fibers, and the Si-avalanche detector efficiency, the
total collection and detection probability of a photon is about 10%.
Consider now the case that two pairs are generated by a single UV-pulse,
and that the four photons are all detected, one by each detector T, D1, D2,
and D3. Our claim is that by the coincident detection of four photons and
because of the brief duration of the UV pulse and the narrowness of the
filters, one can conclude that a three-photon GHZ state has been recorded
5.2. EXPERIMENTAL SET-UP 67
Figure 5.1: Schematic drawing of the experimental setup for the demonstra-tion of Greenberger-Horne-Zeilinger entanglement for three spatially sepa-rated photons. The UV pulse incident on the BBO crystal produces two pairsof entangled photons. Conditioned on the registration of one photon at thetrigger detector T, the three photons registered at D1, D2, and D3 exhibit thedesired GHZ correlations.
68 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT
by detectors D1, D2, and D3. The reasoning is as follows. When a four-
fold coincidence recording is obtained, one photon in path a must have been
horizontally polarized and detected by the trigger detector T. Its companion
photon in path b must then be vertically polarized, and it has a 50% chance
to be transmitted by the beam splitter (see Fig. 5.1) towards detector D3
and a 50% chance to be reflected by the beam splitter towards the final
polarizing beam splitter where it will be reflected to D2. Consider the first
possibility, i.e. the companion of the photon detected at T is detected by
D3 and necessarily carried polarization V . Then the counts at detectors D1
and D2 were due to a second pair, one photon traveling via path a and the
other one via path b. The photon traveling via path a must necessarily be V
polarized in order to be reflected by the polarizing beam splitter in path a;
thus its companion, taking path b, must be H polarized and after reflection
at the beam splitter in path b (with a 50% probability) it will be transmitted
by the final polarizing beam splitter and arrive at detector D1. The photon
detected by D2 therefore must be H polarized since it came via path a and
had to transit the last polarizing beam splitter. Note that this latter photon
was V polarized but after passing the λ/2 plate it became polarized at 45◦
which gave it a 50% chance to arrive as an H polarized photon at detector
D2. Thus we conclude that if the photon detected by D3 is the companion of
the T photon, then the coincidence detection by D1, D2, and D3 corresponds
to the detection of the state
|H〉1 |H〉2 |V 〉3 . (5.2)
By a similar argument one can show that if the photon detected by D2 is
the companion of the T photon, the coincidence detection by D1, D2, and D3
corresponds to the detection of the state
|V 〉1 |V 〉2 |H〉3 . (5.3)
In general, the two possible states (5.2) and (5.3) corresponding to a four-
fold coincidence recording will not form a coherent superposition, i.e. a GHZ
5.2. EXPERIMENTAL SET-UP 69
state, because they could, in principle, be distinguishable. Besides possible
lack of mode overlap at the detectors, the exact detection time of each photon
can reveal which state is present. For example, state (5.2) is identified by
noting that T and D3, or D1 and D2, fire nearly simultaneously. To erase
this information it is necessary that the coherence time of the photons is
substantially longer than the duration of the UV pulse (approximately 200
fs) [Zukowski et al., 1995]. We achieved this by detecting the photons behind
narrow band-width filters which yield a coherence time of approximately 500
fs. Thus, the possibility to distinguish between states (5.2) and (5.3) is no
longer present, and, by a basic rule of quantum mechanics, the state detected
by a coincidence recording of D1, D2, and D3, conditioned on the trigger T,
is the quantum superposition
1√2(|H〉1 |H〉2 |V 〉3 + | V 〉1 |V 〉2 |H〉3) , (5.4)
which is a GHZ state 1
The plus sign in Eq. (5.4) follows from the following more formal deriva-
tion. Consider two down-conversions producing the product state
1
2(|H〉a |V 〉b − | V 〉a |H〉b)
(|H〉′a |V 〉
′b − |V 〉
′a |H〉
′b
). (5.5)
Initially we assume that the components |H〉a,b and |V 〉a,b created in one
down-conversion might be distinguishable from the components |H〉′a,b and|V 〉′a,b created in the other one. The evolution of the individual componentsof state (5.5) through the apparatus towards the detectors T, D1, D2, and
D3 is given by
|H〉a → |H〉T , (5.6)1Rigorously speaking, this erasure technique is perfect, hence produces a pure GHZ
state, only in the limit of infinitesimal pulse duration and infinitesimal filter bandwidth,but detailed calculations [Zukowski et al., 1995; Horne, 1998] reveal that our pulse andfilter values are sufficient to create a clearly observable entanglement, as confirmed by ourexperimental data.
70 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT
|V 〉a →1√2(|V 〉1 + |H〉2) , (5.7)
|H〉b →1√2(|H〉1 + |H〉3) , (5.8)
| V 〉b →1√2(|V 〉2 + |V 〉3) . (5.9)
Identical expressions hold for the primed components. Inserting these ex-
pressions into state (5.5) and restricting ourselves to those terms where only
one photon is found in each output we obtain, after normalization
12
{|H〉T
(|V 〉′1 | V 〉2 |H〉
′3 + |H〉
′1 |H〉
′2 | V 〉3
)+ |H〉′T
(|V 〉1 |V 〉
′2 |H〉3 + |H〉1 |H〉2 |V 〉
′3
)} . (5.10)
If now the experiment is performed such that the photon states from the two
down-conversions are indistinguishable, we finally obtain the desired state
(up to an overall minus sign)
1√2|H〉T (|H〉1 |H〉2 |V 〉3 + |V 〉1 |V 〉2 |H〉3) . (5.11)
Note that, even conditioned upon trigger T detecting a single photon,
the total state of the remaining three photons before detection still contains
terms in which, for example, two photons enter the same detector. Thus,
the GHZ entanglement is observed only under the condition that both the
trigger photon and the three entangled photons are actually detected, and in
the following experiments the four-fold coincidence detection actually plays
the double role of both projecting into the desired GHZ state (5.11) and
performing a specific measurement on the state. This, we submit, in practice
will not be a severe limitation because, on the one hand, in any realistic
scheme one always has losses, and information is only obtained if the photons
are actually observed, as, for instance, in third-man quantum cryptography.
On the other hand, many applications explicitly use specific measurement
results. For example, as we will show in chapter 6, the GHZ argument
5.2. EXPERIMENTAL SET-UP 71
for testing local realism is based on detection events, and knowledge of the
underlying quantum state is actually not even necessary.
The efficiency for one UV pump pulse to yield such a four-fold coincidence
detection is very low (of the order of 10−10). Fortunately, 7.6×107 UV-pulses
are generated per second, which yields about one double pair creation and
detection per 150 seconds, which is just enough to perform our experiments2. Triple pair creations can be completely neglected since they can give rise
to a four-fold coincidence detection of only very few per day.
Comparing with the scheme suggested in Chapter 2 ( refer to Fig. 2.7 ),
the scheme presented here has a significant advantage for the experimental
alignment. That is that one can easily scan into the region of time overlap of
the photons at the final polarizing beamsplitter simply by observing bunching
effect of two correlated photons generated from a single pump pulse. In
contrast, in the scheme of Fig. 2.7 one has to observe bunching effect of
two photons created by independent sources to find the region of quantum
superposition, this correspondingly decreases the two-fold coincidence rate
by at least three orders of magnitude and thus one has to take much longer
scanning time to see clear two-photon bunching effect.
It is also worth noting that one could simply use a 50:50 beam splitter
instead of the polarizing beam splitter in front of the detector T and take
off the half wave plate λ/2 to observe conditional four-photon GHZ entan-
glement. Following the same reasoning line above, one can easily verify that
the state detected by a four-fold coincidence recording of D1, D2, D3 and T
is in the superposition
1√2(|H〉T |V 〉1 | V 〉2 |H〉3 + |V 〉T |H〉1 |H〉2 | V 〉3) ,
which is a four-photon GHZ state.
2The singles detection rate at detectors D1, D2 and D3, is about 15,000 counts persecond, and at the trigger detector T about 100,000 counts per second, due to the largerfilter bandwith and mode acceptance. Four-fold coincidence is registered with logic ANDcircuitry with a coincidence time of 6 ns.
72 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT
5.3 Observation of three-photon entanglement
To experimentally demonstrate that a GHZ state has been obtained by the
method described above, we first verified that, conditioned on a photon de-
tection by the trigger T, both the H1H2V3 and the V1V2H3 components
can be observed with the same intensity, but no others. This was done by
comparing the count rates of the eight possible combinations of polarization
measurements, H1H2H3, H1H2V3, ..., V1V2V3. The observed intensity ratio
between the desired and the undesired states was 12:1. Existence of the two
terms as just demonstrated is a necessary but not yet sufficient condition for
demonstrating GHZ entanglement. In fact, there could in principle be just
a statistical mixture of those two states. Therefore, one has to prove that
the two terms coherently superpose. This we did by a measurement of linear
polarization of photon 1 along +45◦, bisecting the H and V direction. Such
a measurement projects photon 1 into the superposition
|+45◦〉1 =1√2(|H〉1 + | V 〉1) , (5.12)
what implies that the state (5.11) is projected into
1√2|H〉T |+45◦〉1 (|H〉2 | V 〉3 + | V 〉2 |H〉3) . (5.13)
Thus photon 2 and 3 end up entangled as predicted under the notion
of ”entangled entanglement” [Krenn and Zeilinger, 1996]. We conclude that
demonstrating the entanglement between photon 2 and 3 confirms the coher-
ent superposition in state (5.11) and thus the existence of the GHZ entangle-
ment. In order to proceed to our experimental demonstration we represent
the entangled state of photons 2 and 3 in a linear basis rotated by 45◦. The
state then becomes
1√2(|+45◦〉2 |+45◦〉3 − |−45◦〉2 | −45◦〉3) , (5.14)
5.3. OBSERVATION OF THREE-PHOTON ENTANGLEMENT 73
which implies that if photon 2 is found to be polarized along -45◦ (or +45◦),
photon 3 is also polarized along the same direction. We test this predic-
tion in our experiment. The absence of the terms |+45◦〉2 | −45◦〉3 and
| −45◦〉2 |+45◦〉3 is due to destructive interference and thus indicates the
desired coherent superposition of the terms in the GHZ state (5.11). The
experiment therefore consisted of measuring four-fold coincidences between
the detector T, detector 1 behind a +45◦ polarizer, detector 2 behind a -45◦
polarizer, and measuring photon 3 behind either a +45◦ polarizer or a -45◦
polarizer. In the experiment, the difference of arrival time of the photons
at the final polarizer, or more specifically, at the detectors D1 and D2, was
varied.
The data points in Fig. 5.2(a) are the experimental results obtained for
the polarization analysis of the photon at D3, conditioned on the trigger and
the detection of two photons polarized at 45◦ and −45◦ by the two detectorsD1 and D2, respectively. The two curves show the four-fold coincidences for a
polarizer oriented at −45◦ (squares) and +45◦ (circles) in front of detector D3
as function of the spatial delay in path a. From the two curves it follows that
for zero delay the polarization of the photon at D3 is oriented along −45◦,in accordance with the quantum-mechanical predictions for the GHZ state.
For non-zero delay, the photons traveling via path a towards the second po-
larizing beam splitter and those traveling via path b become distinguishable.
Therefore increasing the magnitude of delay gradually destroys the quantum
superposition in the three-particle state.
Note that one can equally well conclude from the data that at zero delay,
the photons at D1 and D3 have been projected onto a two-particle entangled
state by the projection of the photon at D2 onto −45◦. The two conclusionsare only compatible for a genuine GHZ state. We note that the observed
visibility was as high as 75%. 3
3The limited visibility is due mainly to the finite width of the interference filters, thefinite pulse duration, and the limited quality of the polarization optics. detector noise oraccidental coincidences do not play any role.
74 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT
Figure 5.2: Experimental confirmation of GHZ entanglement. Graph (a)shows the results obtained for polarization analysis of the photon at D3, con-ditioned on detection of the trigger photon and detection of one photon atD1 polarized at 45
◦ and one photon at detector D2 polarized −45◦. The twocurves show the four-fold coincidence rates for a polarizer oriented at −45◦and 45◦ respectively in front of detector D3 as a function of the spatial delayin path a. The difference between the two curves at zero delay confirms theGHZ entanglement. By comparison (graph (b)) no such intensity differenceoccurs if the polarizer in front of detector D1 is set at 0
◦. Error bars aregiven by the square root of the coincidence counts.
5.4. DISCUSSION AND CONCLUSION 75
For an additional confirmation of state (5.11) we performed measurements
conditioned on the detection of the photon at D1 under 0◦ polarization (i.e.
V polarization). For the GHZ state (1/√2)(H1H2V3 + V1V2H3) this implies
that the remaining two photons should be in the state V2H3 which cannot give
rise to any correlation between these two photons in the 45◦ detection basis.
The experimental results of these measurement are presented in Fig. 5.2(b).
The data clearly indicate the absence of two-photon correlations and thereby
confirm our claim of the observation of GHZ entanglement between three
spatially separated photons.
5.4 Discussion and conclusion
Although the extension from two to three entangled particles might seem
to be only a modest step forward, the implications are rather profound.
First of all, GHZ entanglements allow for novel tests of quantum mechanics
versus local realistic models [Greenberger et al., 1989; 1990; Mermin, 1990a;
1990b; Zukowski, 1998; Pan et al., 1999a]. Secondly, three-particle GHZ
states might find a direct application, for example, in third-man quantum
cryptography. And thirdly, the method developed to obtain three-particle
entanglement from a source of pairs of entangled particles can be extended to
obtain entanglement between many more particles [Bose et al., 1998], which
is the basis of many quantum communication and computation protocols.
Finally, we note that our experiment, together with our earlier realization
of quantum teleportation [Bouwmeester, Pan et al., 1997] and entanglement
swapping [Pan et al., 1998b] provides the necessary tools to implement a
number of novel entanglement distribution and network ideas as recently
proposed [Grover, 1997; Duer et al., 1999].
Chapter 6
Experimental tests of the GHZtheorem
6.1 Introduction
Ever since its introduction by Schrodinger [Schrodinger, 1935] entanglement
has commanded a central position in the discussions of the interpretation of
quantum mechanics. Originally that discussion has focused on the proposal
by Einstein, Podolsky and Rosen (EPR) of measurements performed on two
spatially separated entangled particles [Einstein et al., 1935]. Most signifi-
cantly, John Bell then showed that there is a conflict between any attempt to
explain the correlations observed in such systems by a local realistic model
and the predictions made by quantum mechanics [Bell, 1964]. In the deriva-
tion of Bell’s inequalities one makes the seemingly innocuous assumption
that perfect correlations can be understood using such a local realistic model
and the conflict then arises for the statistical predictions of quantum theory.
An increasing number of experiments on entangled particle pairs having
confirmed the statistical predictions of quantum mechanics [Freedman et al.,
1972; Aspect et al., 1982; Weihs 1998] have thus provided increasing evidence
against local realistic theories. Yet, one might find some comfort in the fact
76
6.2. THE CONFLICT WITH LOCAL REALISM 77
that such a realistic and thus classical picture can explain perfect correlations
and is only in conflict with statistical predictions of the theory. After all,
quantum mechanics is statistical in its core structure. In other words, for
entangled particle pairs the cases where the result of a measurement on one
particle can definitely be predicted on the basis of a measurement result on
the other particle can be explained by a local realistic model. It is only
that subset of statistical correlations where the measurement results on one
particle can only be predicted with a certain probability which cannot be
explained by such a model.
Yet in 1989 it was shown by Greenberger, Horne and Zeilinger (GHZ) that
for certain three- and four-particle states [Greenberger et al., 1989; 1990] a
conflict with local realism arises even for perfect correlations. That is, even
for those cases where, based on measurement on N − 1 of the particles,
the result of the measurement on particle N can be predicted with certainty.
Local realism and quantum mechanics here both make definite but completely
opposite predictions. A particularly elegant demonstration of that conflict is
due to Mermin [Mermin, 1990a].
Utilizing our recently developed source for three-photon GHZ-entanglements
it is the purpose of this chapter to present a first realization of such a three-
particle test against local realism [Pan et al., 1999a].
6.2 The conflict with local realism
6.2.1 GHZ theorem
How are the quantum predictions of a three-photon GHZ-state in stronger
conflict with local realism than the conflict for two-photon states as implied
78 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM
by Bell’s inequalities 1? To answer this, consider the state
1√2(|H〉1 |H〉2 |H〉3 + | V 〉1 |V 〉2 |V 〉3) , (6.1)
where H and V denote horizontal and vertical linear polarizations. This
state indicates that the three photons are in a quantum superposition of
the state |H〉1 |H〉2 |H〉3 (the three photons are horizontally polarized) and|V 〉1 | V 〉2 |V 〉3 (the three photons are vertically polarized). We choose this
specific state because it is symmetric with respect to the interchange of all
photons which simplifies the arguments below. The same line of reasoning
holds, however, for any three-particle entangled state.
Consider now some specific predictions following from state (6.1) for mea-
surements of linear polarization along directions rotated by 45◦ with respect
to the original H-V directions, denoted by H ′-V ′, or of circular polarization
denoted by L-R (left-handed, right-handed). These new polarizations can be
expressed in terms of the original ones as
|H ′〉 = 1√2(|H〉+ |V 〉), |V ′〉 = 1√
2(|H〉 − |V 〉) ,(6.2)
|R〉 = 1√2(|H〉+ i |V 〉), |L〉 = 1√
2(|H〉 − i |V 〉) .(6.3)
For convenience we will refer to a measurement of H ′-V ′ linear polarization
as a l measurement and of L-R circular polarization as a c measurement.
Representing state (6.1) in the new states using Eqs. (6.2) and (6.3)
one obtains predictions for measurements of these new polarizations. For
example, for the case of measurement of circular polarization on, say, both
photon 1 and 2, and measurement of linear polarization H ′-V ′ on photon 3,
1For two-photon states Hardy [Hardy, 1993] has found situations where quantum me-chanics predicts a specific result to occur sometimes and local realism predicts the sameresult never to occur [Boschi et al., 1997]
6.2. THE CONFLICT WITH LOCAL REALISM 79
denoted as a ccl experiment, the state may be expressed as
12(|R〉1 |L〉2 |H ′〉3 + |L〉1 |R〉2 |H ′〉3+ |R〉1 |R〉2 |V ′〉3 + |L〉1 |L〉2 | V ′〉3)
(6.4)
This expression has a number of significant implications. Firstly, we note
that any specific result obtained in any individual or in any two-photon
joint measurement is maximally random. For example, photon 1 will exhibit
polarization R or L with the same probability of 50%, or photons 1 and 2
will exhibit polarizations RL, LR, RR, or LL with the same probability of
25%.
Yet secondly we realize that, given any two results of measurements on
any two photons, we can predict with certainty what the result of the corre-
sponding measurement performed on the third photon will be. For example,
suppose photons 1 and 2 are both found to exhibit right-handed (R) circular
polarization. Then by the third term in the expression above, photon 3 will
definitely be V ′ polarized.
By cyclic permutation, we can obtain analogous expressions for any case
of any experiment measuring circular polarization on two photons and H ′-V ′
linear polarization on the remaining one. Thus, in any one of the three ccl,
clc and lcc experiments any individual measurement result both for circular
polarization and for linear H ′-V ′ polarization can be predicted with certainty
for any one of the three photons given the corresponding measurement results
of the other two.
Now we will analyze the implications of these predictions from the point
of view of local realism. First note that the predictions are independent of the
spatial separation of the photons and independent of the relative time order
of the measurements. Let us thus consider the experiment to be performed
such that the three measurements are performed simultaneously in a given
reference frame, say, for conceptual simplicity, in the reference frame of the
source. Thus we can employ the notion of Einstein locality which implies
that no information can travel faster than the speed of light. Hence the
80 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM
specific measurement result obtained for any photon must not depend on
which specific measurement is performed simultaneously on the other two nor
on the outcome of these measurements. The only way then to explain from a
local realist point of view the perfect correlations discussed above is to assume
that each photon carries elements of reality for both l and c measurements
considered and that these elements of reality determine the specific individual
measurement result [Greenberger et al., 1989; 1990; Mermin, 1990a].
We now consider a fourth experiment measuring linear H ′-V ′ polariza-
tion on all three photons, i.e. a lll experiment. What possible outcomes
will a local realist predict here based on the elements of reality introduced
to explain the earlier ccl, clc and lcc experiments? The state of Eqn. (6.4)
and its permutations imply that whenever we obtain the result H ′ [V ′] for
any one photon, the other two photons must carry elements of reality imply-
ing opposite [identical] circular polarizations. Suppose that for one specific
run of the lll experiment we find, say, the result V ′ both for photon 2 and
for photon 3. Because photon 3 is a V ′, both photon 1 and 2 must carry
identical circular polarization elements of reality; and because photon 2 is a
V ′, both photons 1 and 3 must carry identical circular polarization elements
of reality. This means that all three photons must carry identical circular
polarization elements of reality. Thus, since photons 2 and 3 carry identical
circular polarization elements of reality, photon 1 must necessarily exhibit
linear polarization V ′, it cannot be H ′ polarized. Hence the existence of ele-
ments of reality leads to the conclusion that the result V ′1V′
2V′
3 is one possible
outcome and H ′1V′
2V′
3 is impossible. By parallel constructions, one can verify
that H ′1H′2V′
3 , H ′1V′
2H′3, and V ′1H
′2H′3 are the only other possible outcomes
from a local realistic viewpoint if we elect to measure H ′-V ′ polarizations of
all three particles, i.e. if we perform a lll measurement.
How do these predictions of local realism compare with those of quan-
tum physics? If we express the state given in Eq. (6.1) in terms of H ′-V ′
6.2. THE CONFLICT WITH LOCAL REALISM 81
polarization using Eq. (6.2) we obtain
12(|H ′〉1 |H ′〉2 |H ′〉3 + |H ′〉1 | V ′〉2 | V ′〉3+ |V ′〉1 |H ′〉2 | V ′〉3 + |V ′〉1 |V ′〉2 |H ′〉3) .
(6.5)
Here the local realistic model predicts none of the terms occurring in the
quantum prediction. This implies that whenever local realism predicts a
specific result definitely to occur for a measurement on one of the photons
based on the results for the other two, quantum physics definitely predicts
the opposite result. For example, if two photons are both found to be H ′
polarized, local realism predicts the third photon to carry polarization V ′
while the quantum state predicts H ′.
Thus, while in the case of Bell’s inequalities for two photons the difference
between local realism and quantum physics happens for statistical predictions
of the theory, for three entangled particles the difference occurs already for
the definite predictions, statistics is now only due to inevitable measurement
errors occurring in any and every experiment, even in classical physics.
6.2.2 Generalization to conditional GHZ state
The experiment reported here is based on the observation of three-photon
GHZ entanglement that was achieved in Chapter 5. Conditioned upon that
detector T observes a single photon, the total photon state out of our setup
(Fig. 5.1) actually reads2
|H〉1|V 〉1|V 〉2 + |H〉1|V 〉1|H〉3+|H〉2|V 〉2|H〉1 + |H〉2|V 〉2|V 〉3+|H〉3|V 〉3|V 〉1 + |H〉3|V 〉3|H〉2+|H〉1|H〉2|H〉3 + |V 〉1|V 〉2|V 〉3
(6.6)
2For simplicity of argumentation we have assumed here that for photon 3 H and V aredefined at right angles compared to photons 1 and 2, and also we already exclude the casein which only one EPR pair is produced by a single pulse or two photons enter detectorT. Yet we note that such a simplification does not change the physical conclusion.
82 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM
which implies that the total photon state produced by our setup, i.e., the state
before detection, also contains terms in which two photons enter the same
detection station. For example, the first two terms of Eq. 6.6 imply there
are two photons in mode 1, and so on. Therefore, the four-fold coincidence
detection here acts as a projection measurement onto the desired GHZ state
(6.1) and filters out those undesirable terms. This might raise doubts about
whether such a source can be used to test local realism.
Actually, the same doubts had been raised earlier for certain Bell-type ex-
periments involving photon pairs [Ou et al., 1988; Shih and Alley, 1988]. Al-
though these experiments have successfully produced the expected quantum-
mechanical correlations, in the past it was often believed [Kwiat et al., 1994,
Caro and Garuccio, 1994] that they could never, not even in their idealized
versions, be consider as genuine tests of local realism. To avoid this problem,
a possible solution is that one could directly prepare unconditional three-
photon GHZ state using the scheme suggested in Chapter 2 (refer to Fig.
2.7).
However, fortunately, Popescu, Hardy and Zukowski [Popescu et al., 1997]
showed that this general belief is wrong and that the above experiments
indeed constitute (modulo the usual detection loopholes) true tests of local
realism. Following the same reasoning line, Zukowski has recently shown
that our GHZ entanglement source enables one to perform a three-particle
test of local realism [Zukowski 1999]. Here we briefly discuss this analysis.
For clarity of the argumentation, let’s first define a local hidden-variable
model for our GHZ entanglement source. In such a model the local events
(i.e., events at one of the observation stations) are determined by the value
of the hidden variables describing the experiment, usually denoted by the
symbol λ, and the local macroscopic controllable parameter set by the local
observer; here the settings of the polarizers in front of detectors D1, D2,
and D3, respectively denoted by x1, x2, and x3. As mentioned already, in
the GHZ argument xi(i = 1, 2, 3) could independently indicate settings for
linear H ′ − V ′ polarization or for circular R − V polarization, i.e. l or c
6.2. THE CONFLICT WITH LOCAL REALISM 83
measurements respectively.
Let Λ be the space of the parameter λ for an ensemble comprised of a
very large number of the states produced by our GHZ entanglement source.
From Eq. 6.6, it is easy to see that such an ensemble can be partitioned into
two disjoint sub-ensembles Λa and Λb, i.e. corresponding to those for which
(a) one photon each is detected in each of the outgoing beams 1, 2 and 3,
(b) two photons are observed in one of the three outgoing beams, and one
photon is observed in one of the remaining two beams. Denoting by ρ(λ)
the distribution of the hidden variable λ, then according to local realism, the
distribution ρ(λ) of the union of the two subsets is clearly independent of
the settings of the polarizers in front of detectors D1, D2, and D3. However,
it is not evident whether the mode of partitioning is also independent of the
settings of the polarizers. If one wants to construct the GHZ argument within
the sub-ensemble Λa, one must thus demonstrate that the distribution ρa(λ)
of the sub-ensemble Λa is also independent of the polarizer settings. For a
thorough discussion of the essence of that argument, refer to [Clauser and
Shimony, 1978] and references therein.
To do so, imagine that detectors D1 and D2 each detects a single photon
(under two specified polarizer settings, x1, x2 respectively), then there is
definitely a single photon to be detected by D3, no matter what the polarizer
setting is there3. This implies that, (1) the occurrence of such a joint event,
i.e. whether a joint event belongs to the subset Λa or not, is irrespective of
the local polarizer setting of D3; (2) the local occurrence of a single photon
detection by D3, which belongs to the subset Λa, is also independent of the
polarizer setting of D3.
By parallel reasoning, one can finally conclude that the occurrence of a
joint event belonging to the sub-ensemble Λa depends only on the hidden
variable λ and not upon the local settings of the polarizers. Furthermore, in
3 Because the four photons were produced by a single pulse, then conditioned upon thatdetectors D1, D2 and T each detects a single photon, detector D3 must also correspondinglydetect a single photon. Here, we always suppose the detectors have perfect 100% detectionefficiency.
84 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM
the subset the occurrence of a single photon detection at any local observation
station also does not depend on the polarizer settings at D1, D2, and D3.
These exactly confirm that the distribution ρa(λ) of the sub-ensemble Λa
is independent of the polarizer orientations, by which we could define the
probability measures only on the subset Λa and take ρa(λ) to have norm one:
∫Λa
dρa = 1 (6.7)
Thus, by limiting ourselves only to the sub-ensemble Λa we can predict the
possible outcomes in a lll experiment via local realism, then compare with the
quantum mechanical prediction. Therefore, we finally arrive at the following
conclusion:
In essence, the GHZ argument for testing local realism is based on de-
tection events, and knowledge of the underlying quantum state is not even
necessary. It is indeed enough to consider only those four-fold coincidences
discussed above and ignore totally the contributions by the other terms.
6.3 Experimental results
As explained in section 6.2.1 demonstration of the conflict between local
realism and quantum mechanics for GHZ entanglement consists of four ex-
periments each with three spatially separated polarization measurements.
First, one performs ccl, clc, and lcc experiments. If the results obtained are
in agreement with the predictions for a GHZ state then the predictions for
an lll experiment are exactly opposite for a local realist theory as to that of
quantum mechanics.
For each experiment we have 8 possible outcomes of which ideally 4 should
never occur. Obviously, no experiment neither in classical physics nor in
quantum mechanics can ever be perfect and therefore, due to principally un-
avoidable experimental errors, even the outcomes which should not occur will
6.3. EXPERIMENTAL RESULTS 85
occur with some small probability in any realistic experiment. The question
is how to deal with this problem in view of the fact that the GHZ argument
is based on perfect correlations.
In the present chapter we follow two independent possible strategies. In
the first strategy we simply compare our experimental results with the pre-
dictions both of quantum mechanics and of a local realist theory for GHZ
correlations assuming that the particles carry the hidden variables necessary
to explain the perfect quantum ccl, clc and lcc correlations. The spurious
events are then just due to experimental imperfection not correlated to the
hidden variables a photon carries. A local realist might argue against that
approach and suggest that the non-perfect detection events indicate that the
GHZ argumentation cannot succeed. In our second strategy we therefore
give maximum leeway to local realist theories assuming that the non-perfect
events in the first three experiments indicate a set of hidden variables (el-
ements of reality) which are in extreme conflict with quantum mechanics.
We then compare the local realist prediction for the lll experiment obtained
under that assumption with the experimental results.
In the experiment, all measurements are conditioned upon the detection
of a photon by the trigger detector T and we will only refer to the remaining
three photons which are detected by detectors D1, D2, and D3. To meet the
condition of time overlap of the photons at the final polarizing beamsplitter
PBS (Fig. 5.1) we change in small steps the time difference between the pho-
tons from arm a and b by translating the position of the beamsplitter BS.
In this way, we can scan into the region of quantum superposition. Polar-
izers and λ/4 plates have been used to perform polarization analysis. More
specifically, we insert a polarizer oriented at 45◦ or -45◦ in front of a certain
detector to perform a H ′ or V ′ polarization measurement respectively, and
further insert a λ/4 plate in front of the polarizer to perform a R or L circular
polarization measurement.
The observed results for two possible outcomes in a ccl experiment are
shown in Fig. 6.1(a). The remaining possible outcomes of a ccl experiment
86 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM
have also been measured. At large delay, i.e. outside the region of coherent
superposition, it was observed that within the experimental accuracy the
eight possible outcomes have the same coincidence rate, whose mean value
was chosen as a normalization standard. After normalizing we determined
the fractions for all eight possible outcomes simply by dividing the normal-
ized four-fold coincidences of a specific outcome by the sum of all possible
outcomes in a ccl experiment. For instance the two black bars in Fig. 6.1(b)
indicate the fractions of the RRV ′ and RRH ′ contributions at zero delay
respectively.
All individual fractions which were obtained in our ccl, clc and lcc exper-
iments are shown in Figs. 6.2(a), (b) and (c), respectively. From the data we
conclude that we observe the GHZ terms of Eq. 6.4 predicted by quantum
mechanics in 85% of all cases and in 15% we observe spurious events.
Adopting our first strategy we assume the spurious events are just due to
experimental errors and thus conclude within the experimental accuracy that
for each photon 1, 2 and 3, quantities corresponding to both c and l mea-
surements are elements of reality. Consequently a local realist if he accepts
that reasoning would thus predict that for a lll experiment, the combinations
V ′V ′V ′, H ′H ′V ′, H ′V ′H ′, and V ′H ′H ′ will only be observable (Fig. 6.3(b)).
However referring back to our original discussion we see that quantum me-
chanics predicts the exact opposite terms should be observed (Fig. 6.3(a)).
To settle this conflict we then perform the actual lll experiment. Our results,
shown in Fig. 6.3(c), disagree with the local realism predictions and are con-
sistent with the quantum mechanical predictions. The individual fractions
in Fig. 6.3(c) clearly show within our experimental uncertainty that only
those triple coincidences predicted by quantum mechanics occur and not
those predicted by local realism. In this sense, we claim that we experimen-
tally realized the first three-particle test of local realism following the GHZ
argument.
We have already seen that the observed results for a lll experiment con-
firm the quantum mechanical predictions when we assume that deviations
6.3. EXPERIMENTAL RESULTS 87
Figure 6.1: A typical experimental result used in the GHZ argument, in thiscase four-fold coincidences (top) between the trigger detector T, detectors D1
and D2 both set to measure a right-handed polarized photon, and detector D3
set to measure a linearly polarized H ′ (lower) and V ′ (upper curve) photon asa function of the delay between photon 1 and 2 at the final polarizing beam-splitter. At zero delay maximal GHZ entanglement results and the bottomgraph shows the experimentally determined fractions of RRV ′ and RRH ′
triples (out of the eight possible outcomes in the ccl experiment) as deducedfor the zero delay measurements
88 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM
Figure 6.2: Fractions of the various outcomes observed in the ccl, clc, andlcc experiments. The experimental data show that we observe the GHZ termspredicted by quantum physics in (85± 4)% of all cases and in (15± 2)% the
6.3. EXPERIMENTAL RESULTS 89
Figure 6.3: The conflicting predictions of quantum physics (a) and local real-ism (b) of the fractions of the various outcomes in a lll experiment for perfectcorrelations. The experimental results (c) are in agreement with quantumphysics within experimental errors and in disagreement with local realism.
90 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM
from perfect correlations in our experiment, and in any experiment for that
matter, are just due to unavoidable experimental errors. However what are
the predictions of local realism for the lll experiment when the correlations
are not perfect, as is the case for our experiment where not all events observed
in the ccl, clc and lcc experiments agree with the quantum GHZ predictions?
Is it possible that by using a local realistic theory, these non-GHZ terms can
explain all our experimental results?
To answer this we adopt our second strategy and consider the best predic-
tion that a local realistic theory could obtain using these spurious terms. How
could, for example, a local realist obtain the quantum prediction H ′H ′H ′?
One possibility is to assume that triple events producing H ′H ′H ′ would be
described by a specific set of local hidden variables such that they would give
events in agreement with quantum theory both in a lcc and clc experiment,
for example the results H ′LR and LH ′R, but a spurious event for a ccl ex-
periment, namely LLH ′. In this way any local realistic prediction for an
event predicted by quantum theory in our lll experiment will use at least one
spurious event in the earlier measurements together with two correct ones.
Therefore the fraction of quantum predictions in the lll experiment can at
most be equal to the sum of the fractions of all spurious events in the ccl,
clc, and lcc experiments, that is 0.45. However, we experimentally observed
such terms with a fraction of 0.87± 0.04 (Fig. 6.3(c)), in clear contradictionto the hidden variable prediction.
6.4 Discussion and Prospects
Since the first tests of quantum mechanics versus local realism there have
been strong debates as to what extent these experiments fully refute the
notion of local realism. In this chapter we presented the first experimental
test of quantum nonlocality in three-particle entanglement where the theories
make definite but opposite predictions. Our experiment fully confirms the
predictions of quantum mechanics and is in conflict with local hidden variable
6.4. DISCUSSION AND PROSPECTS 91
theories. We would like to remark that our second analysis presented above,
succeeds because our average visibility of (71 ± 4)% clearly surpasses the
minimum of 50% necessary for a violation of local realism [Mermin, 1990b;
Roy and Singh, 1991; Zukowski and Kaszlikowski, 1997; Ryff, 1997].
However, we have by no means the illusion that our new test will once
and for all convince the disbelievers of quantum mechanics. Our experiment
shares with all existing two-particle tests of local realism the property that
the detection efficiencies are rather low. Therefore we had to invoke the fair
sampling hypothesis [Pearle, 1970; Clauser and Shimony, 1978] where it is
assumed that the registered events are a faithful representative of the whole
ensemble.
It will be interesting to further study GHZ correlations over large dis-
tances with space-like separated randomly switched measurements [Weihs
et al., 1998], to extend the techniques used here to the observation of multi-
photon entanglement [Bose et al., 1998], to observe GHZ-correlations in mas-
sive objects like atoms [Hagley et al., 1997], and to investigate possible ap-
plications in quantum computation and quantum communication protocols
[Briegel et al., 1998; Cleve and Buhrman, 1997].
Chapter 7
Conclusions and outlook
In this work, we have used pairs of polarization-entangled photons as pro-
duced by pulsed parametric down-conversion to experimentally explore in-
terference phenomena of multiparticle quantum systems. Our research has
been mainly concentrated on the experimental demonstration of quantum
teleportation, on the experimental realization of entanglement swapping, on
the production of three-particle GHZ entanglement, and on the experimental
realization of a three-particle test of local reality versus quantum mechan-
ics. For the first time, these experiments open the door to study various
novel phenomena for quantum systems of three or more particles. It is fore-
seen that the techniques developed in our experiments, besides their interest
to the foundations of quantum physics, will have many important applica-
tions in future quantum communication schemes, such as third-man quantum
cryptography, entanglement purification, and entanglement distribution.
Although teleportation has been realized using polarization-entangled
photons, we are still on the way to long-distance quantum teleportation,
i.e. transmission of quantum states over large distance. Utilizing the tools
that were recently developed for those long-distance Bell-type experiments
[Weihs et al., 1998; Tittel et al., 1998a; 1998b], a slight modification of our
teleportation scheme will allow us to realize long-distance teleportation with
an efficiency of 50%. As discussed in Chapter 3, a stable visibility better than
92
93
71% is necessary to violate a Bell inequality. It is of great interest to inves-
tigate the possibility to improve the visibility of our entanglement swapping
experiment, this will ultimately leads to an experimental test of nonlocality
both with a pair of photons that never interacted and truly independent ob-
servers. It is proposed that one can produce four-photon GHZ state [Pan
and Zeilinger, 1999b]. The long-distance GHZ experiment will constitute
a test for local realism under strict Einstein locality condition. It is also
suggested to perform long-distance quantum cryptography based on multi-
particle (GHZ) entanglement, which enables a more advanced cryptography
system.
We have seen in the thesis that our interferometric Bell-state analyzer
does not give the full capacity of the new quantum communication schemes.
Three instead of four messages can be encoded in one photon and the tele-
portation of polarization states of photon can be performed only with a
maximum of 50% efficiency. To perform all possible unitary transforma-
tions strong coupling between quantum systems is necessary. We propose
to continue the development of photon-photon coupling in optical systems.
As suggested by Roch et al. [Roch et al., 1992], the experiments recently
performed by Kimble’s group at Caltech [Turchette et al., 1995, Hood et
al., 1998] show a way to increase the coupling between weak light beams
by atoms in high finesse cavities. However, the bandwidth requirements of
these devices are too high to combine this approach with down-conversion
experiments.
It will be a real future challenge to study similar systems with slightly
relaxed bandwidth requirements, in order to adapt the technique for our
experiments. This would ultimately lead to the realization of complete long-
distance teleportation of quantum states of photon and atom. Moreover,
this would also open a wholly new field of experimental investigations, since
such photon-photon-coupling devices are needed for quantum nondemolition
measurements, quantum logic gates, and in various experiments on the foun-
dations of quantum mechanics.
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