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Chain rules for quantum channels

Mario Berta, Marco Tomamichel

TL;DR

The paper addresses deriving chain rules for quantum channels in Rényi divergences, extending classical results to the quantum setting. It employs a direct matrix-analysis approach based on spectral pinching and Matsumoto constructions to relate input divergences to channel outputs. The main contribution is a simple meta-chain rule for all alpha>0, along with geometric and sandwiched variants and regularized forms, plus pre-processing techniques that can remove certain complexities. The work clarifies amortized channel divergences and shows that adaptive input strategies do not yield asymptotic advantages in channel discrimination, providing a versatile toolkit for quantum entropy inequalities.

Abstract

Divergence chain rules for channels relate the divergence of a pair of channel inputs to the divergence of the corresponding channel outputs. An important special case of such a rule is the data-processing inequality, which tells us that if the same channel is applied to both inputs then the divergence cannot increase. Based on direct matrix analysis methods, we derive several Rényi divergence chain rules for channels in the quantum setting. Our results simplify and in some cases generalise previous derivations in the literature.

Chain rules for quantum channels

TL;DR

The paper addresses deriving chain rules for quantum channels in Rényi divergences, extending classical results to the quantum setting. It employs a direct matrix-analysis approach based on spectral pinching and Matsumoto constructions to relate input divergences to channel outputs. The main contribution is a simple meta-chain rule for all alpha>0, along with geometric and sandwiched variants and regularized forms, plus pre-processing techniques that can remove certain complexities. The work clarifies amortized channel divergences and shows that adaptive input strategies do not yield asymptotic advantages in channel discrimination, providing a versatile toolkit for quantum entropy inequalities.

Abstract

Divergence chain rules for channels relate the divergence of a pair of channel inputs to the divergence of the corresponding channel outputs. An important special case of such a rule is the data-processing inequality, which tells us that if the same channel is applied to both inputs then the divergence cannot increase. Based on direct matrix analysis methods, we derive several Rényi divergence chain rules for channels in the quantum setting. Our results simplify and in some cases generalise previous derivations in the literature.
Paper Structure (5 sections, 5 theorems, 42 equations)

This paper contains 5 sections, 5 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Notation and methods
  3. Main results
  4. Applications
  5. Outlook

Key Result

Lemma 1

Let $\rho,\sigma\in\mathcal{S}(\mathcal{H})$ and $\alpha\in(0,2]$. Then, we have where the infimum is over probability distributions $P,Q$ over finite alphabets $\mathcal{X}$, and positive trace preserving maps with $\Gamma(P)=\rho$ and $\Gamma(Q)=\sigma$.

Theorems & Definitions (8)

  • Lemma 1: Matsumoto matsumoto10matsumoto14
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Corollary 4
  • Corollary 5
  • proof : Proof of Corollary \ref{['cor:sandwiched-chain']}