Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximate recoverability and the quantum data processing inequality

Saptak Bhattacharya

TL;DR

This work addresses refinements of the quantum data processing inequality in the regime of approximate recoverability. It develops and uses log-convexity properties for θ-divergences and sandwiched Rényi entropies to analyze recoverability under the Petz map, including a counterexample that disproves a prior monotonicity conjecture. The authors prove uniform approximate-recoverability bounds for the sandwiched Rényi entropy $S_{2}$ and refine them to depend only on the base state $B$ and system dimension, then extend the analysis to $S_{2}$-based fidelity bounds and conditional expectations. These results clarify the limits of recoverability via the Petz map and provide practically meaningful, dimension- and state-dependent constants for DPI-type inequalities in quantum information theory.

Abstract

In this paper, we discuss the quantum data processing inequality and its refinements that are physically meaningful in the context of approximate recoverability. An important conjecture regarding this due to Seshadreesan et. al. in J. Phys. A: Math. Theor. 48 (2015) is disproved. We prove some inequalities capturing universal approximate recoverability with the Petz recovery map for the sandwiched quasi and Rényi relative entropies for the parameter $t=2$. We also obtain convexity theorems on some parametrized versions of the relative entropy and fidelity, which can be of independent interest.

Approximate recoverability and the quantum data processing inequality

TL;DR

This work addresses refinements of the quantum data processing inequality in the regime of approximate recoverability. It develops and uses log-convexity properties for θ-divergences and sandwiched Rényi entropies to analyze recoverability under the Petz map, including a counterexample that disproves a prior monotonicity conjecture. The authors prove uniform approximate-recoverability bounds for the sandwiched Rényi entropy and refine them to depend only on the base state and system dimension, then extend the analysis to -based fidelity bounds and conditional expectations. These results clarify the limits of recoverability via the Petz map and provide practically meaningful, dimension- and state-dependent constants for DPI-type inequalities in quantum information theory.

Abstract

In this paper, we discuss the quantum data processing inequality and its refinements that are physically meaningful in the context of approximate recoverability. An important conjecture regarding this due to Seshadreesan et. al. in J. Phys. A: Math. Theor. 48 (2015) is disproved. We prove some inequalities capturing universal approximate recoverability with the Petz recovery map for the sandwiched quasi and Rényi relative entropies for the parameter . We also obtain convexity theorems on some parametrized versions of the relative entropy and fidelity, which can be of independent interest.
Paper Structure (4 sections, 20 theorems, 144 equations)

This paper contains 4 sections, 20 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. The $\theta$-divergence
  3. Sandwiched Rényi entropy, relative entropy difference and recoverability
  4. Approximate recovery with the Petz map

Key Result

Theorem 1

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. Let $x\in\mathcal{A}$ be positive and invertible. Then for every positive definite state $\phi:\mathcal{A}\to\mathbb{C}$ the function $\theta\to\phi(x^{\theta})$ is log-convex on $[0,1]$.

Theorems & Definitions (40)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Lemma 1
  • Theorem 2
  • proof
  • Corollary 3
  • ...and 30 more