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Recoverability in quantum information theory

Mark M. Wilde

arXiv:1505.04661v5quant-phcond-mat.stat-mechcs.IThep-thmath-phmath.ITmath.MP

TL;DR

Quantum relative entropy cannot increase under physical evolutions, and this inequality underlies results in quantum information theory and other areas of physics. The paper proves a refinement using a Rényi relative-entropy difference and recovery maps; small entropy decrease allows one state to be recovered perfectly and the other approximately. It also derives improvements to entropy inequalities, while some questions about the Rényi quantities remain open.

  • Problem

    Quantum relative-entropy monotonicity is foundational to quantum information theory, but physically meaningful refinements of this inequality are needed.

    Source: §1 · p. 2

  • Method

    The paper proves a relative-entropy refinement using complex interpolation, linear algebra, and a Rényi generalization of a relative-entropy difference.

    Source: §1 · p. 2, §2 · p. 3

  • Results

    Small relative-entropy decrease permits a recovery operation that perfectly recovers one state while approximately recovering the other.

    Source: §1 · p. 2, §3 · p. 8

  • Takeaways & Limitations

    The refinement supplies physically meaningful improvements to entropy inequalities, including the strong-subadditivity refinement of [FR14] as a corollary.

    Source: §1 · p. 2

    Whether t can be set to zero and whether the Rényi quantities are monotone non-decreasing in α remain open questions.

    Source: §6 · p. 19

Abstract

from arXiv · show

The fact that the quantum relative entropy is non-increasing with respect to quantum physical evolutions lies at the core of many optimality theorems in quantum information theory and has applications in other areas of physics. In this work, we establish improvements of this entropy inequality in the form of physically meaningful remainder terms. One of the main results can be summarized informally as follows: if the decrease in quantum relative entropy between two quantum states after a quantum physical evolution is relatively small, then it is possible to perform a recovery operation, such that one can perfectly recover one state while approximately recovering the other. This can be interpreted as quantifying how well one can reverse a quantum physical evolution. Our proof method is elementary, relying on the method of complex interpolation, basic linear algebra, and the recently introduced Renyi generalization of a relative entropy difference. The theorem has a number of applications in quantum information theory, which have to do with providing physically meaningful improvements to many known entropy inequalities.

1 Introduction

The paper refines quantum relative-entropy monotonicity with a recovery operation that perfectly recovers one state and approximately recovers another when the entropy decrease is small. It builds on prior exact and approximate recovery results and uses an elementary proof strategy.

1 Introduction

  • Quantum relative-entropy monotonicity and strong subadditivity are prominent entropy inequalities and are equivalent to each other.Entropy inequalities constrain possible physical evolutions and communication efficiencies, and also have applications across physics [Wil13] [BHN+15] [AMPS13].Source: §1 · p. 1
  • Petz showed that a channel acting on two given states can be perfectly reversed iff relative-entropy decrease equals zero, and gave an explicit recovery-channel construction.Source: §1 · p. 1
  • Near-saturation of strong subadditivity implies an approximate quantum Markov chain, allowing one system to be recovered from another while preserving correlations with a third [FR14].The refinement has applications in quantum information theory and is expected to find further applications in physics [LW14] [SBW14] [SW14] [Wil14].Source: §1 · p. 2
  • Small relative-entropy decrease permits a recovery operation that perfectly recovers one state while approximately recovering the other.The proof uses complex interpolation, basic linear algebra, and a Rényi generalization of a relative-entropy difference.Source: §1 · p. 2
  • The paper establishes physically meaningful refinements of quantum relative-entropy monotonicity and derives the strong-subadditivity refinement of [FR14] as a corollary.Whether the converse implication holds, or whether the more general refinement of [SFR15] follows, remains open; the recovery channel also has functoriality properties [LW14].Source: §1 · p. 2

2 Background

The background introduces finite-dimensional quantum systems, operator norms, and information measures used in the proof. Its main technical tool is an operator version of the Hadamard three-line theorem, a variant of the Riesz-Thorin interpolation theorem.

2 Background

  • The paper restricts its developments to finite-dimensional Hilbert spaces, while noting that extension to separable Hilbert spaces should be possible.Density operators represent quantum states, and quantum channels are completely positive trace-preserving maps.Source: §2 · p. 2
  • The Schatten p-norm is a technical tool, and its unitary invariance is repeatedly used in the main proof.The infinity norm is defined as the largest singular value, the limit of the p-norm as p tends to infinity.Source: §2 · p. 3
  • The paper uses a Rényi generalization of a relative-entropy difference and notes that Rényi conditional mutual information is a special case [SBW14] [BSW15a].Information measures are introduced as needed rather than collected in one place.Source: §2 · p. 3
  • The operator Hadamard three-line theorem is the main technical interpolation tool and is a variant of the Riesz-Thorin operator interpolation theorem [Bei13, Dup15, BL76, RS75].The theorem is stated for a bounded operator-valued map that is holomorphic in the strip's interior and continuous on its boundary.Source: §2 · p. 3

3 Bounds for a difference of quantum relative entropies

The section refines relative-entropy monotonicity by bounding the relative-entropy difference with recovery performance for a rotated Petz map. It establishes exact recovery in the zero-difference case and gives conditions and open questions governing the bounds.

3 Bounds for a difference of quantum relative entropies

  • As α tends to 1, the Rényi relative-entropy difference equals D(ρ∥σ) − D(N(ρ)∥N(σ)).The quantum relative entropy is monotone under quantum channels.Source: §3 · p. 4
  • The upper-bound setting requires positive definite states and channel outputs because the proof uses matrix inverses.The more general lower-bound setting allows a density operator and positive semidefinite operator with supp(ρ) contained in supp(σ).Source: §3 · pp. 4–5
  • The proof uses an isometric extension of the channel and the operator Hadamard three-line theorem, taking limits in the Rényi parameter to obtain the bounds.The proof handles the lower-bound and upper-bound inequalities in separate settings.Source: §3 · pp. 6–8
  • The theorem does not determine an optimal rotation parameter t, which may depend on ρ and be undesirable for approximate quantum error correction.The partial isometry Uω,t preserves the density operator ω.Source: §3 · p. 8
  • Theorem 4 states that, for ρ, σ, and N as in Definition 2, a recovery map perfectly recovers σ from N(σ), while its performance recovering ρ from N(ρ) is limited by (3.18).For upper bounds, the paper uses a more restrictive setting for ρ, σ, and N.Source: §3 · pp. 5, 6, 8
  • For α in (1/2, 1), the Rényi relative-entropy difference is nonnegative, improving on [DW15]; when it is zero, a recovery map perfectly recovers both ρ and σ.The zero condition means the channel is sufficient for the two states.Source: §3 · p. 9

4 Functoriality

For fixed-parameter recovery maps, the paper establishes normalization, parallel composition, and serial composition properties. These specify how recovery behaves for identity, product, and sequential channels.

4 Functoriality

  • The fixed-parameter recovery maps in (3.19) satisfy normalization, parallel composition, and serial composition properties.These are the functoriality properties established in this section.Source: §4 · p. 9

4.1 Normalization

  • When σ is positive definite and N is the identity channel, the recovery map is the identity channel.This is the normalization property.Source: §4.1 · p. 9

4.2 Parallel Composition

  • For tensor-product σ and N, the recovery map respects the same product structure.The paper establishes parallel composition for the recovery maps of the form (3.19).Source: §4.2 · pp. 9–10

4.3 Serial Composition

  • For a channel N = N2 ◦ N1, serial composition recovers from N2 first and then from N1.The map associated with N1 recovers from N1.Source: §4.3 · p. 10

5 Consequences and applications of Theorem 4

Theorem 4 strengthens several entropy inequalities with recovery-related remainder terms. This section applies it to strong subadditivity and concavity of conditional quantum entropy.

5 Consequences and applications of Theorem 4

  • Theorem 4 strengthens strong subadditivity, concavity of conditional entropy, joint convexity, non-negativity of quantum discord, the Holevo bound, and multipartite information inequalities.The paper also notes potential applications to approximate quantum error correction.Source: §5 · p. 10

5.1 Strong Subadditivity

  • Corollary 8 improves strong subadditivity and gives an upper bound on conditional mutual information.Its recovery map is a special case of the Petz recovery map.Source: §5.1 · p. 11
  • For α ∈ (1/2, 1), the conditional-mutual-information inequality improves on the non-negativity result in [BSW15a].The recovery parameter t depends on the global state ρABC, so this map does not possess the universality property discussed in [SFR15].Source: §5.1 · p. 12

5.2 Concavity of conditional quantum entropy

  • Theorem 4 yields an improvement of concavity of conditional quantum entropy for an ensemble of bipartite quantum states.The proof encodes the ensemble in a classical-quantum state and applies Theorem 4.Source: §5.2 · pp. 12–13

5.3 Joint convexity of quantum relative entropy

Theorem 4 gives refinements of joint convexity of quantum relative entropy and non-negativity of quantum discord. In the discord application, the recovery map perfectly recovers the measured subsystem’s marginal state.

5.3 Joint convexity of quantum relative entropy

  • Theorem 4 gives an improvement of joint convexity of quantum relative entropy for ensembles whose distinguishability is compared after loss of the classical label.The resulting corollary uses a recovery map related to a rotated pretty good measurement.Source: §5.3 · pp. 13–14

5.4 Non-negativity of quantum discord

  • The discord refinement bounds the fidelity recovery error by I(A; B)ρ − I(X; B)ω.The map UρA,t ◦ EA perfectly recovers ρA from the measurement output MA→X(ρA).Source: §5.4 · pp. 14–15

5.5 Holevo bound

The section applies the paper’s recovery-based refinements to the Holevo bound and to differences of multipartite information. It reports an affirmative resolution of an open question about the latter.

5.5 Holevo bound

  • For quantum-classical ρAB, the Holevo bound compares mutual information before and after measuring A, with the former never smaller.Source: §5.5 · p. 16
  • Applying Corollary 15 and (3.10) yields an improvement to the measurement bound, stated as Corollary 16.Source: §5.5 · p. 16

5.6 Differences of quantum multipartite informations

  • Applying Theorem 4 to differences of multipartite information resolves affirmatively the open question posed in Eq. (7.8) of [Wil14].Source: §5.6 · p. 16
  • For positive-definite states, an additional inequality is stated in the multipartite-information treatment.Source: §5.6 · p. 16

5.7 General quantum information measures

The paper extends its interpolation-based bounds to information quantities that need not be relative-entropy differences, including conditional multipartite information. It also discusses a state-dependence constraint on applying its recovery bound to approximate quantum error correction.

5.7 General quantum information measures

  • Complex interpolation and a Rényi generalization can produce bounds for information quantities that are not expressible as relative-entropy differences, though those bounds need not imply recoverability.Source: §5.7 · p. 17
  • Conditional multipartite information is treated through a Rényi generalization and bounds with a sequential-recoverability interpretation.The procedure involves repeatedly using system C to retrieve the A_i systems one by one.Source: §5.7 · p. 17
  • The conditional multipartite-information bound has no prefactor depending on the number of parties, unlike repeated applications of the chain rule and [FR14].Source: §5.7 · p. 17

5.8 Approximate quantum error correction

  • For approximate quantum error correction, the stated inequality is not particularly useful and motivates further developments.Source: §5.8 · p. 18
  • The recovery map depends on the transmitted state, which the receiver does not know and therefore cannot use to select that decoder.Source: §5.8 · p. 18

6 Discussion

The discussion places the recovery results alongside prior refinements and identifies open questions about the recovery parameter and Rényi quantities. It also points to complex interpolation as a source of simple proofs and possible further applications.

6 Discussion

  • The recovery map in (5.7) lacks the universality property discussed in [SFR15], while the paper’s argument for its input and output unitaries is more direct than [FR14] [SFR15].Source: §6 · p. 19
  • Whether the inequalities hold at t = 0 and whether the Rényi quantities in (3.1) and (5.10) are monotone non-decreasing in α remain open questions.Source: §6 · p. 19
  • The Hadamard three-line theorem yields simple proofs and may have further applications in quantum information theory.Source: §6 · p. 19

A Convergence to the quantum relative entropy difference

This appendix establishes that the Rényi relative-entropy-difference quantity converges to the quantum relative entropy difference. The proof invokes the support inclusion for the states and their channel outputs.

A Convergence to the quantum relative entropy difference

B Petz recovery map

The appendix establishes that the Petz recovery map is linear, completely positive, and trace-non-increasing. Its complete positivity follows from expressing it as a composition of three completely positive maps.

B Petz recovery map

  • The Petz recovery map is linear, completely positive, and trace-non-increasing.The appendix states these properties for the map defined in (3.9).Source: B Petz recovery map · pp. 22–23
  • Complete positivity follows because the map is a composition of three completely positive maps.The composition applies conjugation by [N(σ)]^-1/2, then N†, then conjugation by σ^1/2.Source: B Petz recovery map · p. 23
  • The appendix introduces a trace-non-increasing proof after stating the property.Source: B Petz recovery map · p. 23
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