Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Universal recovery maps and approximate sufficiency of quantum relative entropy

Marius Junge, Renato Renner, David Sutter, Mark M. Wilde, Andreas Winter

TL;DR

This work proves the existence of an explicit universal recovery map $\mathcal{R}_{\sigma,\mathcal{N}}$ that depends only on a reference state $\sigma$ and a quantum channel $\mathcal{N}$, providing a tight, information-theoretic remainder to the data processing inequality: $D(\rho||\sigma) \ge D(\mathcal{N}(\rho)||\mathcal{N}(\sigma)) - 2 \log F\bigl(\rho,(\mathcal{R}_{\sigma,\mathcal{N}}\circ\mathcal{N})(\rho)\bigr)$ for all $\rho$ with $\mathrm{supp}(\rho) \subseteq \mathrm{supp}(\sigma)$. The recovery map is constructed via a universal mixture of rotated Petz maps and satisfies $(\mathcal{R}_{\sigma,\mathcal{N}}\circ\mathcal{N})(\sigma)=\sigma$, establishing a robust criterion for approximate quantum error correction that is not state-dependent. The authors extend the result to infinite dimensions, derive universal refinements for strong subadditivity, concavity of conditional entropy, and joint convexity of relative entropy, and apply the framework to approximate quantum error correction, connecting information-theoretic distinguishability to recoverability with explicit fidelity-based bounds. Overall, the paper provides a powerful, universal tool for evaluating recoverability and robustness of quantum information against noise, with potential practical impact on quantum coding and fault-tolerant protocols.

Abstract

The data processing inequality states that the quantum relative entropy between two states $ρ$ and $σ$ can never increase by applying the same quantum channel $\mathcal{N}$ to both states. This inequality can be strengthened with a remainder term in the form of a distance between $ρ$ and the closest recovered state $(\mathcal{R} \circ \mathcal{N})(ρ)$, where $\mathcal{R}$ is a recovery map with the property that $σ= (\mathcal{R} \circ \mathcal{N})(σ)$. We show the existence of an explicit recovery map that is universal in the sense that it depends only on $σ$ and the quantum channel $\mathcal{N}$ to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.

Universal recovery maps and approximate sufficiency of quantum relative entropy

TL;DR

This work proves the existence of an explicit universal recovery map that depends only on a reference state and a quantum channel , providing a tight, information-theoretic remainder to the data processing inequality: for all with . The recovery map is constructed via a universal mixture of rotated Petz maps and satisfies , establishing a robust criterion for approximate quantum error correction that is not state-dependent. The authors extend the result to infinite dimensions, derive universal refinements for strong subadditivity, concavity of conditional entropy, and joint convexity of relative entropy, and apply the framework to approximate quantum error correction, connecting information-theoretic distinguishability to recoverability with explicit fidelity-based bounds. Overall, the paper provides a powerful, universal tool for evaluating recoverability and robustness of quantum information against noise, with potential practical impact on quantum coding and fault-tolerant protocols.

Abstract

The data processing inequality states that the quantum relative entropy between two states and can never increase by applying the same quantum channel to both states. This inequality can be strengthened with a remainder term in the form of a distance between and the closest recovered state , where is a recovery map with the property that . We show the existence of an explicit recovery map that is universal in the sense that it depends only on and the quantum channel to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.

Paper Structure

This paper contains 9 sections, 11 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Result.
  3. History of the problem.
  4. Main results
  5. Proof of Theorem \ref{['thm_explicit']}
  6. Universal and explicit refinements of other entropy inequalities
  7. Approximate quantum error correction
  8. Conclusion
  9. Proof of Lemma \ref{['thm:op-hirschman']}

Key Result

Theorem 2.1

Let $A$ and $B$ be separable Hilbert spaces. For any $\sigma \in \mathrm{P}(A)$, any $\rho \in \mathrm{S}_{\sigma}(A)$ and any $\mathcal{N} \in \mathrm{TPCP}(A,B)$ we have where the relative entropy and fidelity are defined in eq_defRelEnt and eq_defFidelity, respectively. The recovery map is given by and $\beta_0$ a probability density function on $\mathbb{R}$ defined by The map $\mathcal{P}_{

Figures (1)

  • Figure 1: This plot depicts the probability density $\beta_0$ defined in \ref{['eq_defBeta']} as a function of $t \in�\mathbb{R}$. We see that it is peaked around $t=0$ which corresponds to the Petz recovery map, i.e., $\mathcal{R}^{t=0}_{\sigma, \mathcal{N}} = \mathcal{P}_{\sigma, \mathcal{N}}$.

Theorems & Definitions (23)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4: Functoriality properties
  • Lemma 3.1: SBW14Wilde15
  • Lemma 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • ...and 13 more