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A resource theory of quantum memories and their faithful verification with minimal assumptions

Denis Rosset, Francesco Buscemi, Yeong-Cherng Liang

arXiv:1710.04710v2quant-ph

TL;DR

The paper addresses how to verify and compare memories that preserve quantum information against measure-and-prepare memories. It develops a resource theory based on classically correlated channel transformations and semiquantum signaling games, which completely characterize the channel order and can certify every non-entanglement-breaking channel. The results are limited to finite-dimensional systems.

  • Problem

    Earlier semiquantum tests with a classical second inquiry detect only channels whose Choi operators are steerable, leaving a need for tests that certify all non-entanglement-breaking channels under trusted-input assumptions.

    Source: THE SETUP · p. 3

  • Method

    The paper defines classically correlated supermaps as free channel transformations and uses semiquantum signaling games to construct a complete family of monotones for comparing memories.

    Source: QUANTUM MEMORIES AS RESOURCES · p. 4, SEMIQUANTUM SIGNALING GAMES AND… · p. 7

  • Results

    Every non-entanglement-breaking channel admits a semiquantum signaling game with positive value for that channel and zero value for entanglement-breaking channels.

    Source: SEMIQUANTUM SIGNALING GAMES AND… · p. 7, CONCLUSION · p. 9

  • Takeaways & Limitations

    The framework treats entanglement-breaking channels as free resources and gives faithful tests of quantum memories using trusted inputs.

    Source: CONCLUSION · pp. 8–9

    The results are formulated for finite-dimensional Hilbert spaces; generalization to infinite-dimensional systems remains open.

    Source: CONCLUSION · p. 9

Abstract

from arXiv · show

We provide a complete set of game-theoretic conditions equivalent to the existence of a transformation from one quantum channel into another one, by means of classically correlated pre/post processing maps only. Such conditions naturally induce tests to certify that a quantum memory is capable of storing quantum information, as opposed to memories that can be simulated by measurement and state preparation (corresponding to entanglement-breaking channels). These results are formulated as a resource theory of genuine quantum memories (correlated in time), mirroring the resource theory of entanglement in quantum states (correlated spatially). As the set of conditions is complete, the corresponding tests are faithful, in the sense that any non entanglement-breaking channel can be certified. Moreover, they only require the assumption of trusted inputs, known to be unavoidable for quantum channel verification. As such, the tests we propose are intrinsically different from the usual process tomography, for which the probes of both the input and the output of the channel must be trusted. An explicit construction is provided and shown to be experimentally realizable, even in the presence of arbitrarily strong losses in the memory or detectors.

THE SETUP

The section defines a temporal game in which a referee gives Abby two quantum inputs at different times and uses the resulting correlations to assess quantum-information storage. The second input remains trusted without requiring entangled or classically correlated probes.

THE SETUP

  • The temporal game has one player, Abby, who may retain an arbitrary amount of classical information between inquiries.The setup imposes the same operational assumptions on her at both times.Source: THE SETUP · p. 3
  • Abby receives an unknown state from a known set at t0, then a second unknown state after the waiting period, and finally broadcasts outcome b.The referee knows the density matrices corresponding to the states supplied.Source: THE SETUP · p. 3
  • Repeated input/output correlations p(b|x, y) let the referee assess whether Abby preserves quantum information over δ = t1 − t0.The first input is supplied at t0, the second at t1, and Abby then reports a classical outcome.Source: THE SETUP · p. 3, Figure 3 · p. 3
  • Both inquiries use trusted quantum states, but the referee need not prepare entangled or classically correlated inputs.The preparation device is assumed trusted, reusable, and re-initializable, with independently drawn questions.Source: THE SETUP · p. 3
  • Unlike Ref., which uses a classical second inquiry and detects only channels with steerable Choi operators, this game uses a trusted quantum second input.The text argues this adds no preparation-device assumption beyond the trusted, reusable input preparation already required.Source: THE SETUP · p. 3
  • The paper states that its tests faithfully detect every non-entanglement-breaking channel and are robust to typical loss models.It contrasts this loss robustness with Bell tests.Source: THE SETUP · p. 3

QUANTUM MEMORIES AS RESOURCES

The resource theory treats entanglement-breaking measure-and-prepare channels as free and channels that preserve entanglement as genuine quantum-memory resources. It compares channels through classically correlated supermaps and uses monotones, including semiquantum-game values, to characterize their relative power.

QUANTUM MEMORIES AS RESOURCES

  • Free channels are entanglement-breaking measure-and-prepare maps, with a POVM outcome and shared randomness selecting the output state.These channels are equivalent to ones that produce separable outputs when applied locally to any bipartite state.Source: QUANTUM MEMORIES AS RESOURCES · p. 4
  • Channels outside the free set preserve entanglement for at least some inputs and therefore require quantum memory in their implementation.The theory allows unlimited classical storage but excludes storage of quantum information as a free operation.Source: QUANTUM MEMORIES AS RESOURCES · p. 4
  • Free transformations use pre- and post-channel quantum operations, instruments, and arbitrary classical memory, while preserving the entanglement-breaking set.The post-processing operation may depend on both the stored instrument outcome and preexisting randomness.Source: QUANTUM MEMORIES AS RESOURCES · p. 4
  • The authors choose classically correlated supermaps as free transformations because they respect temporal order while allowing classical memory and can prepare any free channel.They note that narrower choices would impose artificial restrictions on memory users.Source: QUANTUM MEMORIES AS RESOURCES · p. 5
  • The relation N ֌ N′ holds when a free supermap transforms N into N′, defining a transitive partial order in which some channels may be incomparable.The identity channel is the most powerful resource in the theory because post-processing it can simulate any channel.Source: QUANTUM MEMORIES AS RESOURCES · p. 5
  • Monotones cannot increase under free transformations and are constant across free channels; a complete family characterizes the channel order.Semiquantum-game monotones distinguish every channel outside the free set from free resources.Source: QUANTUM MEMORIES AS RESOURCES · p. 5
  • The Schmidt number of a memory equals that of its Choi operator and measures the maximum quantum dimension preserved, but its practical computation is deferred.Absolute and generalized robustness can also be computed in this resource theory.Source: QUANTUM MEMORIES AS RESOURCES · p. 5

SEMIQUANTUM SIGNALING GAMES AND EXISTENCE OF A TEST

Semiquantum signaling games provide a complete way to compare quantum channels under the paper’s free transformations. They also yield tests that certify every non-entanglement-breaking channel using trusted quantum inputs at two times.

SEMIQUANTUM SIGNALING GAMES AND EXISTENCE OF A TEST

  • A game sends Abby quantum states ξx and ψy at two successive times, then scores her answer using a publicly announced payoff function.The referee chooses questions x and y from finite sets; Abby returns an answer from a finite set.Source: SEMIQUANTUM SIGNALING GAMES AND… · p. 6
  • Abby uses the channel once per round and may use unlimited classical memory and free transformations to adapt it before the second quantum question.The output and second question may then be measured jointly.Source: SEMIQUANTUM SIGNALING GAMES AND… · p. 6
  • Any admissible strategy scoring above a game’s entanglement-breaking threshold certifies a non-entanglement-breaking channel, even if it does not maximize payoff.This makes certification robust to imperfect experimental implementations.Source: SEMIQUANTUM SIGNALING GAMES AND… · pp. 6–7
  • If N achieves at least N′’s maximal payoff in every signaling semiquantum game, then N can be transformed into N′ by a free transformation.Conversely, each game’s maximal payoff is a monotone under these transformations.Source: SEMIQUANTUM SIGNALING GAMES AND… · pp. 6–7
  • Every channel outside the entanglement-breaking class has a signaling semiquantum game that distinguishes it from free resources.The game can be chosen so its maximal payoff is positive while the entanglement-breaking threshold is zero.Source: SEMIQUANTUM SIGNALING GAMES AND… · p. 7

CONSTRUCTION OF EXPERIMENTALLY FRIENDLY SEMIQUANTUM SIGNALING GAMES

The paper constructs semiquantum signaling tests from entanglement witnesses and channel signatures. The construction reduces measurement and input requirements, detects qubit depolarizing channels at a stated threshold, and remains faithful under isotropic erasure losses.

CONSTRUCTION OF EXPERIMENTALLY FRIENDLY SEMIQUANTUM SIGNALING GAMES

  • The signature correlation contains full tomographic data about the channel and is used to construct semiquantum games that witness non-entanglement-breaking behavior.The paper characterizes channels using their Choi-Jamiołkowski representation and an entanglement witness.Source: CONSTRUCTION OF EXPERIMENTALLY… · p. 7
  • A non-entanglement-breaking channel has an entangled Choi state, whose entanglement witness can be decomposed into input-state pairs to define a game with positive payoff and zero entanglement-breaking threshold.The construction provides a constructive test for channels in the quantum domain.Source: CONSTRUCTION OF EXPERIMENTALLY… · p. 7
  • The test can replace a complete Bell measurement with a partial one and reduce the required input pairs to d^2 + 3, compared with at least d^4 for tomographically complete input sets.Here d = min(dim HX, dim HY).Source: CONSTRUCTION OF EXPERIMENTALLY… · p. 8
  • The qubit depolarizing test detects channels for ν > 1/3 ≈33%, versus CHSH thresholds of ν > 2^-1/2 ≈71% or ν > 2^-1/4 ≈84% using two copies.The expected payoff is (3ν −1)/4.Source: Example: qubit depolarizing channel · p. 8
  • For isotropic erasure, Eη ◦ N is in the quantum domain if and only if N is, for every η > 0.The construction models losses and detection inefficiencies independent of the input indices as an erasure channel, with erased states identifiable.Source: Robustness against losses · p. 8

CONCLUSION

The paper frames quantum memories as a resource theory of channels preserving entanglement and develops semiquantum games whose payoffs provide a complete family of monotones. The authors identify finite-dimensional scope and the relation between these monotones and operational channel measures as open directions.

CONCLUSION

  • Semiquantum-game maximal payoffs form a complete family of monotones and yield measurement-device-independent tests that certify all memories in the quantum domain, even with arbitrary losses.The resource theory mirrors entanglement theory for states, with classically correlated pre/post-processing transformations.Source: CONCLUSION · p. 9
  • The relation between the channel monotones and measures such as channel capacities or entropic quantities remains an open question.The authors also point to possible bounds on operational entanglement measures, by analogy with spatial semiquantum frameworks.Source: CONCLUSION · p. 9
  • The framework treats space and time on equal footing and is suited to examining quantum causal structures with arbitrary causal orders.The authors state that it avoids issues associated with describing time-like joint states.Source: CONCLUSION · p. 9
  • The results are derived for finite-dimensional memories; generalization to infinite-dimensional systems remains open.The construction uses Bell measurements, whose generalization to continuous degrees of freedom has a different nature.Source: CONCLUSION · p. 9

Use of randomness in the resource theory of

The resource theory permits preexisting randomness in its definitions, but finite-dimensionality lets the authors absorb it into the operations. This yields equivalent free transformations without randomness and establishes convexity of admissible correlations.

Use of randomness in the resource theory of

  • Preexisting randomness supplies convexity in the definitions of entanglement-breaking channels and free transformations, but can be embedded in quantum operations and classical communication.The authors simplify the notation by incorporating this randomness into the operations.Source: Use of randomness in the resource… · p. 12
  • In finite dimensions, every free transformation using randomness has an equivalent form without randomness.The construction folds the randomness index into the instrument outcome.Source: Use of randomness in the resource… · p. 12
  • All entanglement-breaking channels can be obtained from another entanglement-breaking channel by free transformations, so their definition also needs no preexisting randomness.The admissible correlations for a channel in a scenario are convex.Source: Use of randomness in the resource… · p. 12

Proof of Theorem 1

The proof establishes that ordering channels by their maximal payoffs in signaling semiquantum games is equivalent to a free transformation between them. It constructs a signature correlation to recover the target channel and also bounds the number of input pairs needed for witness decompositions.

Proof of Theorem 1

  • If N achieves at least N′’s maximal expected payoff for every payoff function in a fixed signaling semiquantum scenario, N can reproduce all correlations admissible for N′.Convexity of the admissible correlation sets lets the payoff inequalities imply this inclusion.Source: Proof of Theorem 1 · p. 12
  • A channel reproducing the target’s signature correlation can be transformed into the target by a free transformation.The proof uses tomographically complete inputs and teleportation corrections to recover the target channel.Source: Proof of Theorem 1 · p. 13

Sparse witness decompositions

  • The witness decomposition can be made sparse, with at most d^2 + 3 nonzero input-pair coefficients for d = min(dim HX, dim HY).The cited comparison is with at least d^4 coefficients for full tomographic data.Source: Sparse witness decompositions · p. 14
  • The sparse decomposition method starts from an operator-Schmidt decomposition and rewrites its Hermitian terms using unnormalized density matrices.The terms are shifted to have nonnegative eigenvalues before constructing the decomposition used by the witness.Source: Sparse witness decompositions · p. 14
  • For the qubit depolarizing-channel example, the construction gives a witness decomposition with six nonzero coefficients.The authors note its similarity to the decomposition used in experimental work, while stating their construction is general and not tailored to a specific witness family.Source: Sparse witness decompositions · p. 15
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