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Monotonicity of quantum relative entropy and recoverability

Mario Berta, Marius Lemm, Mark M. Wilde

arXiv:1412.4067v3quant-phcs.ITmath-phmath.ITmath.MP

TL;DR

Relative entropy decreases under quantum operations, and this paper asks how to quantify that loss through recovery. It proves a remainder using a rotated Petz recovery map and shows that weaker Petz-map refinements of equivalent entropy inequalities are equivalent, although whether any holds remains open.

  • Problem

    The paper seeks a remainder for relative-entropy monotonicity using only the Petz recovery map, rather than a rotated version.

    Source: §1 · pp. 2–3

  • Method

    The proof combines methods from [FR14] with relative typical subspaces [BSS03] to derive a rotated Petz recovery-map remainder.

    Source: §1 · p. 3

  • Results

    The rotated Petz remainder quantifies recovery after a quantum operation, while weaker Petz-map refinements of the equivalent inequalities are all equivalent, with their truth still open.

    Source: §1 · pp. 3–4, §5 · p. 13

  • Takeaways & Limitations

    Proving or disproving any one of the weaker Petz-map refinements would settle the others, and the equivalence supports the Petz remainder as a natural candidate.

    Source: §1 · p. 4

    For the partial-trace setting, performance of the rotated Petz map on the reduced state σB remains unquantified, and removing its unitary channels is unresolved.

    Source: §3 · pp. 9–10

Abstract

from arXiv · show

The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, arXiv:1410.0664] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, arXiv:quant-ph/0307170]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.

1 Introduction

The paper refines relative-entropy monotonicity with a remainder term based on a rotated Petz recovery map, combining methods from [FR14] with relative typical subspaces [BSS03]. It also shows that weaker Petz-map refinements of equivalent entropy inequalities stand or fall together, though their truth remains open.

1 Introduction

  • Monotonicity of relative entropy is equivalent to partial-trace monotonicity, joint convexity, strong subadditivity, and concavity of conditional entropy.The paper presents these inequalities as foundational across quantum information theory.Source: §1 · p. 2
  • The paper establishes a similar remainder refinement for monotonicity of relative entropy with respect to partial trace.Source: §1 · p. 3
  • The rotated Petz recovery map remainder quantifies how well ρ can be recovered after a quantum channel, using the Petz map between input and output unitary channels.The proof combines methods from [FR14] with relative typical subspaces from [BSS03, pages 4-5].Source: §1 · p. 3
  • The paper treats refinements using only the Petz recovery map as desirable, but leaves whether they hold as an open question.A weaker proposed form uses the square of the Bures distance on the right-hand side.Source: §1 · pp. 3–4
  • The weaker Petz-map remainder refinements of the equivalent inequalities are themselves equivalent, so they are either all true or all false.The authors say this equivalence reduces proving or disproving the refinements to a single case and supports the Petz remainder as a natural one.Source: §1 · p. 4

2 Relative typical subspace

The section reviews relative typical subspaces and their probability under repeated copies of a state. For fixed tolerance, the corresponding projection captures probability approaching one exponentially fast.

2 Relative typical subspace

  • For fixed δ, the relative typical-subspace projection captures probability at least 1 − ε for sufficiently large n, with convergence to one exponentially fast.The passage attributes the convergence rate to the Hoeffding inequality.Source: §2 · p. 5

3 Remainder term for monotonicity of relative entropy with respect to partial trace

Theorem 1 refines relative-entropy monotonicity under partial trace. Its proof follows [FR14] with modifications and uses relative typical projectors.

3 Remainder term for monotonicity of relative entropy with respect to partial trace

  • Theorem 1 gives a remainder refinement for partial-trace monotonicity when supp(ρ_AB)⊆supp(σ_AB) and σ_B is positive definite.The theorem assumes finite-dimensional systems and defines ρ_B and σ_B by tracing out A.Source: §3 · p. 5
  • The recovery term uses unitary channels U_B and V_AB around the Petz recovery map from B to AB.The proof defines these channels using unitaries and notes that the unitaries can depend on n.Source: §3 · pp. 5, 9
  • The proof follows [FR14] with a few modifications and uses relative typical projectors, the gentle measurement lemma, and trace-norm properties.It also recalls several lemmas from [FR14].Source: §3 · pp. 5–6
  • For density-operator σ_AB, quantifying recovery of σ_B with the rotated Petz map remains open; an alternative completely positive map is not known to be trace preserving.The proposed Petz-only form is conjectured, and the paper states that the alternative map recovers σ_AB perfectly from σ_B while its performance on ρ_AB is bounded by (3.48).Source: §3 · pp. 9–10

4 Remainder term for monotonicity of relative entropy

The section derives a remainder for relative-entropy monotonicity that quantifies recovery using a rotated Petz map, under stated support and positivity conditions.

4 Remainder term for monotonicity of relative entropy

  • The result assumes supp(ρS) ⊆ supp(σS) and that NS→B(σS) is positive definite.Source: §4 · p. 10
  • The proof represents the channel by adjoining an ancilla, applying a unitary, and tracing out part of the output.The Stinespring representation uses a unitary WSE′→BE.Source: §4 · p. 10
  • The proof restricts the optimizing unitary to act isometrically on the relevant support, then extends it to the ancilla-output image.Source: §4 · p. 11
  • The remainder quantifies how well ρ can be recovered from N(ρ) using a rotated Petz recovery map.The section gives the final remainder term for monotonicity of relative entropy.Source: §4 · p. 12
  • For density-operator σS, the performance of the rotated Petz map on NS→B(σS) remains unquantified.Source: §4 · p. 12

5 Equivalence of relative entropy inequalities with remainder terms

The section establishes that refinements of several equivalent entropy inequalities using Petz recovery are themselves equivalent. Whether any of these refinements holds remains open.

5 Equivalence of relative entropy inequalities with remainder terms

  • Theorem 5 makes the entropy inequalities with Petz-recovery remainder terms equivalent, although whether any one is true remains open.The remainder terms are expressed using the square of the Bures distance between density operators.Source: §5 · p. 13
  • The refinements cover strong subadditivity, conditional-entropy concavity, partial-trace monotonicity, and joint convexity of relative entropy.Source: §5 · p. 13
  • Figure 1 situates the Petz-recovery refinements as an extension of the known equivalence among the fundamental entropy inequalities.Source: §5 · p. 13, Figure 1 · p. 14
  • The proof uses a linear perturbation σAB + xρAB and evaluates the derivative of conditional entropy as x approaches zero.The argument analyzes the derivative of the fidelity-based remainder at x = 0.Source: §5 · pp. 15–17

A Auxiliary lemmas from [FR14]

The appendix reproduces auxiliary lemmas used from Fawzi and Renner, including results on trace-nonincreasing maps and permutation-invariant operators.

A Auxiliary lemmas from [FR14]

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