Entanglement cost of quantum state preparation and channel ... · Entanglement cost of quantum...
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Entanglement cost of quantum statepreparation and channel simulation
Xin Wang
QuICS, University of Maryland
Joint work with Mark M. Wilde (LSU)arXiv:1809.09592 & 1807.11939
QIP 2019, University of Colorado Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Entanglement and its manipulationI Separable state: ρAB =
∑i piρ
iA ⊗ ρiB .
I Entangled state: ρAB 6=∑
i piρiA ⊗ ρiB .
I The most natural set of free operations for entanglement manipulationconsists of local operations and classical communication (LOCC), whichhas a complex structure [Chitambar et al’14].
I Entangled states cannot be created by LOCC.
I Inspired the resource theory framework: free states + free operations.(Review paper [Chitambar, Gour’18]).
I The seminal ideas coming from it are influencing diverse areas: quantumthermodynamics, quantum coherence and superposition,non-Gaussianity, stabilizer quantum computation.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Quantifying entanglement
I Entanglement is a key physical resource in quantum information,quantum computation and quantum cryptography.
I A quantitative theory is highly desirable to fully exploit the powerof entanglement.
I Entanglement measure EI Faithfulness: E (ρ) = 0 if and only if ρ is separable.
I Monotonicity: E (Λ(ρ)) ≤ E (ρ) for any Λ ∈ LOCC.
I Strong monotonicity, convexity, additivity, etc.I Zoo of entanglement measures [Plenio, Virmani’07; Christandl’s thesis].I Resource measure provides precise and operationally meaningful
ways to quantify a given physical resource.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Zoo of entanglement measures
I Entanglement measures motivated by operational tasks.
I Entanglement cost EC [Bennett et al’96] quantifies the rate r ofconverting Φ⊗rn to ρ⊗n with an arbitrarily high fidelity in the limit of n.
I [Hayden, Horodecki, Terhal’00] proved that EC equals the regularizedentanglement of formation [Bennett, DiVincenzo, Smolin, Wootters’96].
I Distillable entanglement ED quantifies the rate of the reverse task.
Extremely hard to compute (LOCC + asymptotic)!!!
I Efficiently computable measures, e.g., logarithmic negativity [Vidal,Werner’01], Rains bound [Rains’01].
No direct operational meaning.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Question
Open question
Is there any measure E? with efficiently computable formulaand a direct operational meaning?
I If there is, it will make the analysis of entanglement easier.
I Better understand the fundamental properties of entanglement.
I Applications to operational tasks.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Main result: κ-entanglement gives the exact ent. cost
I Yes, there is! The κ-entanglement
Eκ(ρAB) := inflog Tr SAB : −STBAB ≤ ρ
TBAB ≤ STB
AB , SAB ≥ 0.
I Partial transpose: |ij〉〈kl |TBAB = |il〉〈kj |AB .
I Efficiently computable: Eκ can be computed by semidefiniteprogramming.
Direct operational meaning in entanglement dilution
The exact entanglement cost under PPT operations is given by
EPPT (ρAB) = Eκ(ρAB).
I Properties: Monotonicity, Additivity, Normalization, Faithfulness.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Exact entanglement cost
How many copies of Φ(2) do we haveto invest per copy of target ρAB?
I One-shot exact entanglement cost
E(1)Ω (ρAB) = inf
Λ∈Ω
log d : ρAB = ΛAB→AB(Φd
AB),
I Exact entanglement cost: The minimal number of EPR pairs we needto prepare ρ in an asymptotic setting with zero error:
EΩ(ρAB) = lim infn→∞
E(1)Ω (ρ⊗nAB)/n.
I ELOCC is difficult to solve [Terhal, Horodecki’00] and unknown for basicquantum states so far.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Exact entanglement cost under PPT operations
I Positive-partial transpose (PPT)ρTB ≥ 0 (Peres–Horodecki criterion for separability).
I The most common set of quantum operations beyond LOCC consistsof PPT operations (TB′ ΛAB→A′B′ TB is also completely positive).
I Applications in distillation, quantum communication, etc.I When PPT operations are free, previous bounds for EPPT were
established in [Audenaert, Plenio, and Eisert’03].I Only tight for certain states such as the two-qubit states [Ishizaka’04].
I In general, EPPT(ρ) ≤ ELOCC(ρ).
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Result 1: exact entanglement cost
Exact entanglement cost under PPT operations
For any state ρAB , we have
EPPT (ρAB) = Eκ(ρAB),
where Eκ(ρAB) = inflog Tr S ,−STB ≤ ρTBAB ≤ STB , S ≥ 0.
I Obtain one-shot characterization (symmetry+Choi matrix+PPT)[Audenaert, Plenio, and Eisert’03; Matthews, Winter’08].
E(1)PPT(ρAB) = inf
σ∈S(A⊗B)
log2 m : − (m − 1)σTB
AB ≤ ρTB
AB ≤ (m + 1)σTB
AB
.
I Note that E (1)PPT(ρAB) is not an SDP (bilinear constraints).
I Difficulty: non-convex optimization+ regularization: EPPT(ρAB) = limn→∞ E
(1)PPT(ρ⊗nAB)/n.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Our strategy
I We find that Eκ gives an SDP sandwiched approximation:
I Then the additivity (via SDP duality theory) leads to
EPPT(ρ) ≤ lim infn→∞
1n
log(2Eκ(ρ⊗n) + 1)
≤ lim infn→∞
1n
log(2nEκ(ρ) + 1) (additivity of Eκ)
= Eκ(ρ).
I Similarly, EPPT(ρ) ≥ Eκ(ρ). Thus, EPPT(ρ) = Eκ(ρ).Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Applications of Eκ
I Recalling that EPPT = Eκ, the additivity of Eκ implies theadditivity of exact entanglement cost under PPT operations.
I Exact entanglement cost violates convexity:
∃ρ1, ρ2, EPPT((ρ1 + ρ2)/2) > (EPPT(ρ1) + EPPT(ρ2))/2.
I Exact entanglement cost violates monogamy inequalityI If two qubits A and B are maximally quantumly correlated they
cannot be correlated at all with a third qubit C.I Coffman-Kundu-Wootters (CKW) monogamy inequality:
E (ρAB) + E (ρAC ) ≤ E (ρA(BC)),
where the entanglement in E (ρA(BC)) is understood to be withrespect to the bipartite cut between systems A and BC .
I Concurrence, squashed entanglement satisfies CKW inequality.I For the tripartite state |ψ〉ABC = 1
2 (|000〉+ |011〉+√2|110〉),
EPPT(ψAB) + EPPT(ψAC ) > EPPT(ψA(BC)).
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Entanglement costin quantum channel simulation
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Channel simulationResource trade-off
I classical communication(in protocol F)
I shared entanglement Φ
I Toy model: quantum teleportation [Bennett et al.’93], one ebit in Φ andtwo classical bits in F to exactly simulate a qubit noiseless channel.
I When ent. is free, the classical bits required to simulate a channel inthe asymptotic regime is given by the quantum reverse Shannontheorem [Bennett, Devetak, Harrow, Shor, Winter’14]:
I When classical communication is free, [Berta, Brandao, Christandl,Wehner’11] introduced the entanglement cost of a quantum channel.
EC (N ) := inflog r : limn→∞
infF∈LOCC
‖N⊗n −F(· ⊗ Φ(2rn))‖ = 0
And they proved that
EC (N ) = limn→∞
maxψn
EF (N⊗n ⊗ I(ψn))/n.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Exact entanglement cost of channel simulation
I One-shot exact entanglement cost ofNA→B , under Ω operations
E(1)Ω (N ) = inf
Λ∈Ω
log d : N = ΛA0B0A→B(· ⊗ Φd
A0B0).
I Exact parallel entanglement cost ofNA→B under Ω operations
E(p)Ω (N ) = lim inf
n→∞
1nE
(1)Ω (N⊗n).
I When PPT operations are free, we obtain
E(1)PPT(N ) = inf logm
s.t.− (m − 1)QTB
AB ≤ (JNAB)TB ≤ (m + 1)QTB
AB ,
QAB ≥ 0, TrB QAB = 1A
I E(1)PPT(N ) is not a convex optimization, which makes E (p)
PPT(N ) intractable.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Result 2: entanglement cost of parallel channel simulation
κ-entanglement of a quantum channelWe define the κ-entanglement of a quantum channel NA→B as
Eκ(N ) = inflog ‖TrB QAB‖∞ : −QTB
AB ≤ (JNAB)TB ≤ QTB
AB , QAB ≥ 0,
where JNAB is the Choi operator of N .
I Similar one-shot SDP sandwiched approximation:
log(2Eκ(N ) − 1) ≤ E(1)PPT(NA→B) ≤ log(2Eκ(N ) + 1).
I Apply the SDP duality to get the additivity of Eκ(N ).
Entanglement cost of a quantum channelFor a quantum channel NA→B , the exact parallel entanglement cost ofNA→B is equal to its κ-entanglement:
E(p)PPT(NA→B) = Eκ(NA→B).
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Properties of κ-entanglement of a channel
I A surprising property of Eκ(NA→B) is that
Eκ(NA→B) = supρRA
Eκ(NA→B(ρRA)),
where the supremum is with respect to all ρRA with system R arbitrary.
I As a central quantity, Eκ(N ) has the following fundamental properties:1. Monotonicity under PPT superchannels;2. Additivity Eκ(N1 ⊗N2) = Eκ(N1) + Eκ(N2);3. Normalization: Eκ(Id) = log d ;4. Faithfulness: Eκ(N ) = 0 iff N is PPT;5. Amortization inequality:
Eκ(NA→B(ρA′AB′))− Eκ(ρA′AB′) ≤ Eκ(NA→B).
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Sequential vs. Parallel channel simulation
I An (n,M) exact sequential simulation code consists of a resourcestate ΦM
A0B0and a set P i
AiAi−1B i−1→BiAiB ini=1 of free operations.
I The main idea behind sequential channel simulation is to simulate nuses of the channel NA→B in such a way that they can be called in anarbitrary order, i.e., on demand when they are needed.
I Compatible with a discrimination strategy that can test the the abovesimulation in a sequential way [Chiribella et al’09; Gutoski’12].
I It is the most general paradigm for quantum channel simulation.I Sequential channel simulation is stronger than parallel simulation, thus
has a higher resource cost.Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Result 3: exact entanglement cost of sequential simulation
Exact entanglement cost of sequential simulation
For any quantum channel NA→B , the sequential exact entangle-ment cost is given by
EPPT(NA→B) = Eκ(NA→B).
I Our key contribution is the following sandwiched approximation
log[2nEκ(N ) − 1
]≤ EPPT(NA→B , n) ≤ log
[2(n+1)Eκ(N ) − 1
2Eκ(N ) − 1
].
I Lower bound: sequential simulation cost ≥ parallel simulation costI Achievable part: A protocol that forces the resource after every
round to be maximally entangled and reuses it.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Result 4: resource-seizable channel and entanglement cost
I Resource-seizable channel: Let NA→B be a teleportation-simulablechannel with associated resource state ωA′B′ .Then the channel NA→B is resource-seizable if there exists a separableinput state ρAMABM
to the channel and a postprocessing LOCC channelDAMBBM→A′B′ such that the resource state ωA′B′ can be seized from thechannel NA→B as follows:
DAMBBM→A′B′(NA→B(ρAMABM)) = ωA′B′ .
Entanglement cost of resource-seizable channel
For a resource-seizable channel with associated resource stateωA′B′ , the sequential/parallel entanglement cost of the channelis equal to the entanglement cost of the resource state ωA′B′ :
EC (N ) = E(p)C (N ) = EC (ωA′B′).
I Quantifies the ebits required to sequentially/parallelly simulatethe channel with a vanishing error in the asymptotic regime.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Applications: entanglement cost of fundamental channelsAs applications, we compute EPPT and EC for fundamental quantumchannels including(1) Erasure channel Ep(ρ) = (1− p)ρ+ p|e〉〈e|:
EC (Eq) = E(p)C (Eq) = (1− q) log d ,
EPPT(Ep) = EPPT(p)(Ep) = log(d(1− p) + p).
(2) Dephasing channels Dq(ρ) = (1− q)ρ+ qZρZ :
EC (Dq) = E(p)C (Dq) = h2
(12
+√q (1− q)
)EPPT(Dq) = EPPT
(p)(Dq) = log(1 + 2|q − 1/2|).
(3) Depolarizing ND,p(ρ) = (1− p)ρ+ pd2−1
∑0≤i,j≤d−1(i,j)6=(0,0)
X iZ jρ(X iZ j)†:
EPPT(ND,p) =
log d(1− p) if 1− p ≥ 1
d
0 if 1− p < 1d
(4) Single-mode bosonic Gaussian channels: we give analytical solutions forEC and EPPT.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Entanglement cost of dephasing channel
I For the dephasing channel Dq(ρ) = (1− q)ρ+ qZρZ ,
EC (Dq) = E(p)C (Dq) = h2
(12
+√
q (1− q)
)EPPT(Dq) = EPPT
(p)(Dq) = log(1 + 2|q − 1/2|).
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Entanglement cost of pure-loss channelI Pure-loss Gaussian channel
Lη : b =√ηa +
√1− ηe,
where transmissivity η ∈ (0, 1), environment in vacuum state, and a, b,and e are the field-mode annihilation operators for the input, the output,and the environment’s input, respectively.
I Entanglement cost of Lη:
EC (Lη) = E(p)C (Lη) = h2(1− η)/(1− η).
I The resource theory of entanglement for pure-loss channel is irreversible.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Summary
I Eκ of a quantum stateI Efficiently computable by semidefinite programming.I Direct operational meaning as the exact entanglement cost of state
preparation under PPT quantum operations.I Eκ of a quantum channel
I Efficiently computable by semidefinite programming.I Gives the exact entanglement cost of sequential/parallel simulation.
I Application 1: understanding the fundamental structure of entanglementI When PPT operations are free, the exact entanglement cost is additive.I Exact entanglement cost violates convexity and monogamy inequalities.
I Application 2: Computes the exact entanglement cost of quantum statesand channels under PPT operations. (Benchmark the case of LOCC).
I For a resource-seizable channel, the sequential/parallel entanglement costis equal to the entanglement cost of the underlying resource states.
I Application 3: Solve the entanglement cost of basic quantum channels.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Outlook
I Is the exact entanglement cost under LOCC (the regularizedlog Schmidt rank) additive?
I For an arbitrary Gaussian channel N described by a scalingmatrix X and a noise matrix Y [Serafini’17], do we have
EPPT(N )=12
log min
(1 + detX )2
detY, 1
? (1)
I Resource theory framework for quantum channels?
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder
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Background Exact entanglement cost Entanglement cost of quantum channels Applications Summary
Thank you for your attention!
See arXiv:1809.09592 & 1807.11939 for further details.
Xin Wang (QuICS, UMD) | Entanglement cost of quantum state preparation and channel simulation | QIP 2019, Boulder