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Universal recovery map for approximate Markov chains

David Sutter, Omar Fawzi, Renato Renner

arXiv:1504.07251v3quant-phcs.ITmath-phmath.ITmath.MP

TL;DR

Known recovery proofs were nonconstructive, leaving open whether one recovery map could work for every extension sharing ρBC. The paper establishes such a universal map, with recovery distance bounded by conditional mutual information, and connects the result to two definitions of topological order.

  • Problem

    Known proofs did not explicitly give a recovery map, and it was unclear whether one could depend only on ρBC rather than the full ρABC.

    Source: §1 · p. 2

  • Method

    The paper proves that for each ρBC there is a recovery map determined by that marginal and applicable to every extension ρABC.

    Source: §1 · p. 2, §2 · p. 3

  • Results

    The universal map bounds the recovery distance for any extension by I(A:C|B).

    Source: §1 · p. 2

  • Takeaways & Limitations

    The result implies a close connection between two approaches to defining topological order in quantum systems.

    Source: §1 · p. 2

    The transpose map's corresponding inequality does not extend to all mixed states, even when they are completely classical.

    Source: §7 · p. 22

Abstract

from arXiv · show

A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations. More precisely, we prove that the conditional mutual information $I(A:C|B)$ of a tripartite quantum state $\rho_{ABC}$ can be bounded from below by its distance to the closest recovered state $\mathcal{R}_{B \to BC}(\rho_{AB})$, where the $C$-part is reconstructed from the $B$-part only and the recovery map $\mathcal{R}_{B \to BC}$ merely depends on $\rho_{BC}$. One particular application of this result implies the equivalence between two different approaches to define topological order in quantum systems.

1 Introduction

The introduction frames approximate Markov chains through conditional mutual information and asks whether recovery can be chosen without knowing the full tripartite state. The main result gives a universal recovery map determined by ρBC, with recovery error bounded by conditional mutual information.

1 Introduction

  • Small conditional mutual information motivates approximate Markov chains, since states with small I(A:C|B) need not be close to a perfect Markov chain.The cited work instead relates small conditional mutual information to approximate satisfaction of the recovery condition.Source: §1 · p. 1
  • Previously, known proofs did not construct the recovery map, and it remained open whether a general map could depend only on ρBC rather than ρABC.Such marginal-dependent maps were known in special cases, including perfect Markov chains and states with classical B.Source: §1 · p. 2

2 Main result

  • For any ρBC, one recovery map depending only on ρBC bounds the recovery distance for every extension ρABC by I(A:C|B).This makes the map universal across extensions sharing the same BC marginal.Source: §1 · p. 2
  • The finite-dimensional theorem admits a tighter statement, and recovery maps of form (10) yield an efficiently computable semidefinite program for maximizing fidelity.Whether the same computation is possible for form (11) is unknown.Source: §2 · p. 3
  • The universal map must send ρB to ρBC, a property previously not known to coexist with the recovery bound.This follows by applying the theorem to a product extension ρA⊗ρBC, whose conditional mutual information is zero.Source: §2 · p. 3
  • For product marginals ρBC=ρB⊗ρC, the unique universal map on the support of ρB is the transpose map, and equality holds for relative entropy.The uniqueness argument uses perfect recovery of a Markov state formed with a purification of ρB.Source: §2 · p. 4

3 Applications

The section connects two definitions of topological quantum order using the universal recovery result. It also frames conditional mutual information as a nonnegative quantity with a universal remainder-term strengthening.

3 Applications

  • Strong subadditivity makes conditional quantum mutual information nonnegative, and the theorem strengthens this inequality with a remainder term depending only on ρBC.The section notes that conditional mutual information is used in entanglement measures, many-body correlations, quantum de Finetti results, and quantum information complexity.Source: §3 · p. 4
  • For regions A, B, and C forming topology F, TQO’ is defined by I(A : C|B)ρ = 2γ > 0.Here γ denotes topological entanglement entropy.Source: §3 · p. 4, Figure 1 · p. 5
  • TQO requires family members to be asymptotically orthogonal and to have asymptotically identical marginals on sufficiently small subregions.The section presents this characterization alongside the TQO’ criterion based on conditional mutual information.Source: §3 · p. 4
  • The universal recovery theorem connects TQO and TQO’, addressing an open question about whether a family exhibiting TQO implies that most members exhibit TQO’.The family’s shared BC marginal lets the theorem provide a recovery map that works for each member; continuity of conditional mutual information is also used.Source: §3 · p. 4

4 Proof for finite dimensions

The finite-dimensional proof turns recovery maps that work separately for each extension into one map that works for all extensions sharing a BC marginal. It applies a minimax-style proposition and then extends the result beyond finite-dimensional A.

4 Proof for finite dimensions

  • The finite-dimensional B and C argument uses the measured-relative-entropy version of the recovery inequality, which the cited result [9] supplies for each extension.The proof verifies that the corresponding function family satisfies the proposition’s required properties before applying it.Source: §4 · pp. 5, 11
  • A proposition converts extension-by-extension recovery guarantees into one recovery map valid for every extension in a set with the same BC marginal.The function family must satisfy four stated properties; the proof applies the proposition to the difference between the two sides of the recovery inequality.Source: §4 · pp. 5–6
  • Induction handles finite sets of extensions, and compactness plus approximation extends the proposition to infinite sets.The proof then uses the proposition to establish the theorem for fixed finite-dimensional A.Source: §4 · pp. 6, 9, 11
  • A sequence of finite-rank restrictions and compactness yields a map independent of the dimension of A.The argument first establishes a map for finite-dimensional A, then embeds any separable A isometrically into ℓ2 and uses invariance under isometries.Source: §4 · pp. 11–12

5 Extension to infinite dimensions

The proof extends the recovery theorem to separable, potentially infinite-dimensional systems by finite-rank projection and approximation. It first constructs a sequence of recovery maps and then obtains a limiting map.

Step 1: Existence of a sequence of recovery maps

  • The extension proof reduces the problem to finite-rank projections on B and C, using the shared BC marginal to construct a sequence of recovery maps.The projected states converge in trace norm, and the proof controls the change in the recovery inequality as the projections expand.Source: §5 · p. 12, Step 1: Existence of a sequence of… · pp. 12, 14
  • The sequence has recovery performance approaching the required bound for every extension with the fixed BC marginal.The proof uses finite-dimensional recovery maps on projected systems and a vanishing error term tied to the common marginal.Source: Step 1: Existence of a sequence of… · p. 14

Step 2: Existence of a limit

  • A limit argument produces a recovery map for separable B and C, completing the theorem beyond the finite-dimensional case.The proof controls projected maps and their limits, then uses continuity of fidelity to conclude the recovery statement.Source: Step 2: Existence of a limit · pp. 14, 17, 18

6 Proof of Corollary 2.4

The proof constructs a recovery map for Corollary 2.4 by first finding a structured map for a fixed extension system, then making it universal and removing dependence on that system. For finite-dimensional B and C, the resulting map works for any separable extension system A.

Step 1: Structure of a non-universal recovery map

6 Proof of Corollary 2.4

7 Discussion

The discussion distinguishes the paper’s universal recovery result from the conjectured simple transpose or Petz map. It also identifies limits of that map’s guarantee and leaves a relative-entropy strengthening as an open question.

7 Discussion

  • The main result guarantees a recovery map depending only on ρBC for every extension, while the conjectured simple choice is the transpose or Petz recovery map.The conjectured choice is known to work for Markov states and states with classical B.Source: §7 · p. 22
  • The transpose-map inequality with square-root fidelity holds for pure ρABC, but the proposed extension to mixed states is false even for completely classical states.The pure-state statement is derived from Barnum and Knill’s theorem [2].Source: §7 · p. 22
  • Whether measured relative entropy in the lower bound can be strengthened to relative entropy remains open when optimizing over recovery maps.The corresponding inequality is known to be false when the recovery map is restricted to the transpose map.Source: §7 · p. 22
  • If the relative-entropy inequality holds for every state with a corresponding recovery map, the paper’s argument would yield a universal map satisfying that stronger bound.The discussion proposes applying Proposition 4.1 to a function family built from the relative-entropy difference.Source: §7 · p. 22

A General facts about the fidelity

This appendix records fidelity facts used in the paper’s arguments. It gives a mixture concavity property, its equality condition for orthogonal supports, and a reduction used to extend the Fuchs–van de Graaf inequality to non-negative operators.

A General facts about the fidelity

  • Orthogonal supports let the mixture fidelity attain the weighted-average bound exactly.The proof uses an ancilla to distinguish the orthogonal components and purifications to establish equality.Source: A General facts about the fidelity · p. 23
  • Fidelity of mixtures is at least the corresponding weighted average of the component fidelities.Equality holds when the supports of the paired states are mutually orthogonal.Source: A General facts about the fidelity · p. 23
  • The generalized Fuchs–van de Graaf result for non-negative operators is reduced to equal-trace operators and then to normalized states.The proof adds an orthogonal operator to equalize traces, rescales, and applies the state inequality.Source: A General facts about the fidelity · pp. 23–24

B General facts about the measured relative entropy

This appendix defines measured relative entropy and records properties used in the paper’s arguments. The supplied passages describe its measurement-based definition, convexity, and a proof step that combines measurements.

B General facts about the measured relative entropy

  • Measured relative entropy is the supremum, over POVMs, of the relative entropy between the measured inputs.The definition applies to density operators ρ and σ.Source: B General facts about the measured… · p. 24
  • One proof step combines two measurements into a POVM and then invokes the orthogonality lemma.The passages identify the combined POVM’s elements as the union of the two measurement families.Source: B General facts about the measured… · p. 24
  • The measured relative entropy satisfies a convexity property for mixtures of its second argument.The proof uses convexity of relative entropy and takes the supremum over measurements.Source: B General facts about the measured… · p. 25

C Basic topological facts

This appendix establishes compactness of several finite-dimensional operator and map spaces, then continuity of recovery-map operations used in the proofs.

C Basic topological facts

  • Non-negative operators with bounded trace, or with a fixed marginal, form compact sets in finite dimensions.The appendix derives compactness for trace-bounded operators and operators whose marginal is bounded by or equal to a fixed operator.Source: C Basic topological facts · pp. 25–26
  • Trace-non-increasing and trace-preserving completely positive maps between finite-dimensional spaces form compact sets via the Choi-Jamiolkowski representation.The map sets correspond to compact sets of positive operators with partial trace bounded by or equal to the identity.Source: C Basic topological facts · p. 26
  • Applying a fixed completely positive trace-preserving map, including with an identity map on another system, is continuous; fidelity after recovery is continuous as well.The fidelity claim follows from continuity of the recovered state and of fidelity.Source: C Basic topological facts · p. 26

D Touching sets lemma

The touching-sets lemma guarantees arbitrarily close points from two sets that partition the unit interval and contain its endpoints.

D Touching sets lemma

  • For any δ > 0, the two sets contain u and v with 0 ≤ v − u ≤ δ.The proof takes the infimum of the set containing 1 and handles whether that boundary point belongs to the set containing 0.Source: D Touching sets lemma · p. 26

E Properties of projected states

This appendix develops projection-based approximation facts used in the main proof, including convergence of projected states and control of trace-norm errors.

E Properties of projected states

  • The authors first prove a variant of the gentle measurement lemma [44], which they use repeatedly in the proof of Theorem 2.1.Source: E Properties of projected states · p. 26
  • Finite-rank projectors converging weakly to the identity capture asymptotically all of a density operator’s trace.The statement is given for any density operator on a separable Hilbert space.Source: E Properties of projected states · p. 27
  • The approximation argument is framed for bipartite states sharing a fixed G marginal, with projected sets restricting the E marginal to a projector’s support.The sets S and S_e encode these marginal and support conditions.Source: E Properties of projected states · p. 27
  • Projected approximants with the prescribed marginal converge to the target state in trace norm.The text combines the preceding estimate with a triangle-inequality bound; the Fuchs-van de Graaf inequality is cited in establishing convergence.Source: E Properties of projected states · p. 28

F The transpose map is not square-root optimal

The appendix gives a mixed-state counterexample to a bound for the transpose map. The example does not invalidate the main recovery bound, but indicates that guarantees relative to optimal recovery may require a weaker exponent or a different distance.

F The transpose map is not square-root optimal

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