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Quantum conditional mutual information and approximate Markov chains

Omar Fawzi, Renato Renner

arXiv:1410.0664v3quant-ph

TL;DR

For quantum systems, it is less clear how small conditional mutual information characterizes approximate Markov chains, and some states with small values remain far from every perfect Markov chain. The paper constructs a quantum operation that recovers a state with distance from the original bounded by conditional mutual information, also implying strong subadditivity.

  • Problem

    It is unclear how to characterize states with small conditional mutual information in the quantum setting, including cases where such states are far from every Markov chain.

    Source: §1 · p. 1

  • Method

    For any tripartite state, the paper establishes a quantum operation from B to B ⊗ C that reconstructs a state from the AB marginal.

    Source: §1 · p. 2, §5 · p. 15

  • Results

    Conditional mutual information upper-bounds the trace distance between the original state and a recovered state.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The reconstruction inequality implies strong subadditivity of von Neumann entropy.

    Source: §1 · p. 2

    A logarithmic dim(A) term is necessary in general, and whether the recovery map can be the particular Petz form for positive conditional mutual information remains open.

    Source: §1 · pp. 3–4

Abstract

from arXiv · show

A state on a tripartite quantum system $A \otimes B \otimes C$ forms a Markov chain if it can be reconstructed from its marginal on $A \otimes B$ by a quantum operation from $B$ to $B \otimes C$. We show that the quantum conditional mutual information $I(A: C | B)$ of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated.

1 Introduction

The paper asks how conditional mutual information characterizes approximate quantum Markov chains, where the quantum case is less understood than the classical one. It proves that conditional mutual information upper-bounds the distance to a state reconstructed by a quantum operation from B to B ⊗ C.

Background

  • Zero conditional mutual information characterizes states whose C system can be reconstructed by a quantum operation on B; for classical B, the conditional A ⊗ C states are product states.The paper notes that these classical conditional states express independence of A and C given B.Source: §1 · p. 1

Motivation

  • Prior examples have small conditional mutual information yet large distance to every Markov chain [30] [17], motivating a focus on reconstruction accuracy rather than distance to a perfect Markov chain.The paper targets states with conditional mutual information at most ε and proves a variant of a conjecture for a map not necessarily of Petz form.Source: §1 · pp. 1–2

Main result

  • For every tripartite state, a quantum operation from B to B ⊗ C reconstructs a state whose trace distance from the original is upper-bounded by conditional mutual information.The result concerns distance to a reconstructed state, which is not necessarily itself a Markov chain.Source: §1 · pp. 2, 4

Implications

  • The reconstruction inequality implies strong subadditivity, and equality in the associated data-processing inequality characterizes the Markov chain condition.The equality characterization reproduces the result from [28].Source: §1 · p. 2

Limitations

  • A logarithmic dim(A) term is necessary in general, and whether the recovery map can have the particular Petz form when conditional mutual information is positive remains open.The trace distance is at most 1, while conditional mutual information may reach 2 log2 dim A.Source: §1 · pp. 3–4

Applications

  • The result is motivated by device-independent quantum key distribution and a proposed lower bound on squashed entanglement based on distance to k-extendible states.The latter also implies a strong lower bound in terms of distance to the closest separable state.Source: §1 · p. 4

Proof structure

  • The proof combines one-shot entropy bounds on von Neumann relative entropy with an extension of de Finetti reductions and a tool for evaluating fidelity of permutation-invariant states.These techniques are applied successively in two main steps.Source: §1 · p. 4

2 Typicality bounds on the relative entropy

This section develops relative-entropy bounds using one-shot entropy measures and the Asymptotic Equipartition Property. It also establishes stability under trace-nonincreasing maps and derives typicality consequences.

Typicality bounds on the relative entropy

  • One-shot versions of relative entropy, combined with the Asymptotic Equipartition Property, provide bounds on von Neumann relative entropy.The approach parallels deriving strong subadditivity for one-shot entropies before passing to von Neumann entropy.Source: §2 · pp. 4–5
  • If the output trace decreases less than exponentially, applying the same trace-nonincreasing completely positive map to product-state arguments cannot substantially increase their relative entropy.The section notes this stability condition is stronger than an alternative inequality when the trace loss is appreciable.Source: §2 · pp. 6–7
  • The Asymptotic Equipartition Property follows as a corollary of the relative-entropy stability lemma and is used in the proof of Theorem 5.1.The typicality argument establishes exponential decay of both spectral tails.Source: §2 · pp. 7–8

3 Generalised de Finetti reduction

The section extends de Finetti reduction to permutation-invariant bipartite states with a known marginal. It uses this reduction to handle permutation-invariant maps as well as states.

Generalised de Finetti reduction

  • The reduction extends to bipartite states whose D^n marginal is a tensor power, and its proof allows the state to be mixed by using a permutation-invariant purification.The extension represents the relevant symmetric state through a probability measure over extensions.Source: §3 · pp. 9, 11, 12
  • For permutation-invariant states with a known marginal on D^n, the analysis can be reduced to product states built from extensions of the given state on D.The generalized reduction is based on de Finetti methods and uses representation theory in its proof.Source: §3 · p. 9
  • A corresponding reduction applies to permutation-invariant completely positive trace-preserving maps, represented through a probability measure over such maps.The map version follows by applying the inverse Choi–Jamiołkowski isomorphism to the state reduction.Source: §3 · p. 12

4 Fidelity between permutation-invariant operators

This section develops fidelity bounds for permutation-invariant operators using the generalized de Finetti reduction. The bounds replace many-copy comparisons with comparisons involving extensions on a single copy and dimension factors.

Fidelity between permutation-invariant operators

  • The section uses generalized de Finetti reduction and established fidelity properties to approximate fidelity between permutation-invariant states.Its stated purpose is to provide techniques used in the proof of Theorem 5.1.Source: §4 · p. 12
  • For permutation-invariant operators, a suitable extension of a fixed marginal yields a fidelity bound with dimension factor d = dim(D) dim(E)^2.When the operator is pure, the extension can be pure and the factor can be bounded by max[dim(D), dim(E)].Source: §4 · p. 13
  • For permutation-invariant operators on (R ⊗ S)^n, a related fidelity bound uses the dimension factor d = dim(R) dim(S)^2.The lemma allows an additional permutation-invariant operator on R^n.Source: §4 · p. 13

5 Main result and proof

Theorem 5.1 gives a trace-preserving recovery map from B to B ⊗ C for states on separable Hilbert spaces. The proof establishes the claim first in finite dimensions and extends it to separable spaces.

5 Main result and proof

  • Theorem 5.1 guarantees a trace-preserving completely positive map from B to B ⊗ C for any density operator on separable A, B, and C.For finite-dimensional systems, the map has a specified form and can be chosen on the support of ρB.Source: §5 · p. 15
  • The proof establishes the main inequality in finite dimensions, then extends it to separable Hilbert spaces by finite-rank approximation.The extension uses trace-norm convergence of projected states and continuity and monotonicity properties stated in the proof.Source: §5 · pp. 15, 17, 18
  • The proof's limiting argument yields the inequality after the auxiliary positive parameters are allowed to approach zero.The optimized fidelity is independent of n, and the proof invokes this to take the asymptotic limit.Source: §5 · p. 17
  • The recovery map can be chosen to reproduce the B and C marginals exactly.Source: §5 · p. 19

A One-shot relative entropies

Appendix A reviews generalized relative entropy and smooth max-relative entropy, then states bounds and asymptotic properties used in the paper. It also notes a monotonicity property of the smoothed quantity.

A One-shot relative entropies

  • The appendix reviews generalized relative entropy and smooth max-relative entropy, including a slightly modified variant used in the paper.Source: A One-shot relative entropies · p. 19
  • The generalized relative entropy is defined for non-negative operators ρ and σ and smoothing parameter ε in [0, tr(ρ)].Source: A One-shot relative entropies · p. 19
  • The smoothed generalized relative entropy is monotonically non-increasing in the smoothing parameter.Source: A One-shot relative entropies · p. 20
  • The appendix gives an upper bound on smooth max-relative entropy using trace distance to an operator bounded above by a multiple of σ.The cited lemma assumes ρ̄ ≤ λσ for λ > 0.Source: A One-shot relative entropies · p. 20
  • For product states, the Quantum Asymptotic Equipartition Property gives an asymptotic statement for smooth max-relative entropy, and the generalized relative entropy bound holds asymptotically with equality.The latter equality is stated as a consequence of Quantum Stein’s Lemma.Source: A One-shot relative entropies · pp. 20–21

B General facts about the fidelity

Appendix B collects fidelity properties for non-negative operators and completely positive maps. It also gives a permutation-invariant purification result that can be extended to other symmetry groups.

B General facts about the fidelity

C On the Schur-Weyl duality

Appendix C states a Schur-Weyl decomposition and uses it to describe vectors in the symmetric subspace. The construction relies on maximally entangled vectors in paired representation spaces.

C On the Schur-Weyl duality

D Strong faithfulness of squashed entanglement

This section derives a quantitative version of squashed entanglement’s faithfulness by relating it to distance from k-extendible states. The resulting trace-norm bound depends on the dimension of only one subsystem, and an antisymmetric-state example shows that such a dimension factor is necessary.

D Strong faithfulness of squashed entanglement

  • Theorem 5.1 yields a quantitative version of faithfulness by relating squashed entanglement to distance from the closest k-extendible state.Squashed entanglement is strictly positive exactly for entangled states.Source: D Strong faithfulness of squashed… · p. 26
  • The Corollary D.3 trace-norm bound for squashed-entanglement faithfulness depends on the dimension of only one subsystem.By symmetry, the lower-dimensional subsystem can be chosen.Source: D Strong faithfulness of squashed… · p. 28
  • For the totally antisymmetric state, squashed entanglement is O(1/d), while its distance to the closest separable state is at least 1/4.The example shows that a dimension factor in the bound is necessary; any product state has overlap at most 1/2 with the antisymmetric subspace.Source: D Strong faithfulness of squashed… · p. 28
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