Entanglement, quantum phase transitions and quantum algorithms
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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