Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum $f$-divergences and Their Local Behaviour: An Analysis via Relative Expansion Coefficients

Shreyas Iyer, Peixue Wu, Paula Belzig, Graeme Smith

TL;DR

This work analyzes quantum $f$-divergences and their local second-order behaviour through relative expansion coefficients, linking global distinguishability to its local (Riemannian) geometry. It proves a no-reverse-DPI result over all states for non-unitary channels with $d_B\le d_A$, and shows generic, channel-independent equalities between divergence and Riemannian coefficients in two infinite families of $f$-divergences, while introducing an equivalence framework to compare different measures. It demonstrates that, for QC channels, standard and Riemannian coefficients coincide, and it develops an equivalence theory distinguishing bounded vs unbounded kernels, with strict-positivity results for strictly positive channels. The paper then applies these ideas to approximate recoverability and to primitive quantum channels, proving a reverse-quantum-Markov convergence theorem and deriving constructive recovery bounds, and it provides explicit positive coefficients for concrete channels (generalised dephasing, qubit dephasing, and amplitude damping). Overall, the results offer a unifying view of information preservation under quantum channels, with practical implications for recoverability and mixing times in quantum processes and open questions on further equality and boundedness phenomena.

Abstract

Any reasonable measure of distinguishability of quantum states must satisfy a data processing inequality, that is, it must not increase under the action of a quantum channel. We can ask about the proportion of information lost or preserved and this leads us to study contraction and expansion coefficients respectively, which can be combined into a single \emph{relative expansion coefficient}. We focus on two prominent families: (i) standard quantum $f$ divergences and (ii) their local (second-order) behaviour, which induces a monotone Riemannian semi-norm (that is linked to the $χ^2$ divergence). Building on prior work, we identify new families of $f$ for which the global ($f$ divergence) and local (Riemannian) relative expansion coefficients coincide for every pair of channels, and we clarify how exceptional such exact coincidences are. Beyond equality, we introduce an \emph{equivalence} framework that transfers qualitative properties such as strict positivity uniformly across different relative expansion coefficients. Leveraging the link between equality in the data processing inequality (DPI) and channel reversibility, we apply our framework of relative expansion coefficients to approximate recoverability of quantum information. Using our relative expansion results for primitive channels, we prove a reverse quantum Markov convergence theorem, converting positive expansion coefficients into quantitative lower bounds on the convergence rate.

Quantum $f$-divergences and Their Local Behaviour: An Analysis via Relative Expansion Coefficients

TL;DR

This work analyzes quantum -divergences and their local second-order behaviour through relative expansion coefficients, linking global distinguishability to its local (Riemannian) geometry. It proves a no-reverse-DPI result over all states for non-unitary channels with , and shows generic, channel-independent equalities between divergence and Riemannian coefficients in two infinite families of -divergences, while introducing an equivalence framework to compare different measures. It demonstrates that, for QC channels, standard and Riemannian coefficients coincide, and it develops an equivalence theory distinguishing bounded vs unbounded kernels, with strict-positivity results for strictly positive channels. The paper then applies these ideas to approximate recoverability and to primitive quantum channels, proving a reverse-quantum-Markov convergence theorem and deriving constructive recovery bounds, and it provides explicit positive coefficients for concrete channels (generalised dephasing, qubit dephasing, and amplitude damping). Overall, the results offer a unifying view of information preservation under quantum channels, with practical implications for recoverability and mixing times in quantum processes and open questions on further equality and boundedness phenomena.

Abstract

Any reasonable measure of distinguishability of quantum states must satisfy a data processing inequality, that is, it must not increase under the action of a quantum channel. We can ask about the proportion of information lost or preserved and this leads us to study contraction and expansion coefficients respectively, which can be combined into a single \emph{relative expansion coefficient}. We focus on two prominent families: (i) standard quantum divergences and (ii) their local (second-order) behaviour, which induces a monotone Riemannian semi-norm (that is linked to the divergence). Building on prior work, we identify new families of for which the global ( divergence) and local (Riemannian) relative expansion coefficients coincide for every pair of channels, and we clarify how exceptional such exact coincidences are. Beyond equality, we introduce an \emph{equivalence} framework that transfers qualitative properties such as strict positivity uniformly across different relative expansion coefficients. Leveraging the link between equality in the data processing inequality (DPI) and channel reversibility, we apply our framework of relative expansion coefficients to approximate recoverability of quantum information. Using our relative expansion results for primitive channels, we prove a reverse quantum Markov convergence theorem, converting positive expansion coefficients into quantitative lower bounds on the convergence rate.

Paper Structure

This paper contains 22 sections, 29 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. An Introduction to $f$-divergences
  5. Operator Convex Functions
  6. Relative Expansion Coefficients
  7. No Reverse Data Processing Inequality over All States
  8. Equality between Divergence and Riemannian Coefficients
  9. Inequality between Divergence and Riemannian Coefficients
  10. The Special Cases of Generic Equality
  11. Consequence of a Classical Output
  12. Equivalence
  13. The Notion of Equivalence and its Inheritability
  14. Relative Expansion Coefficients in the Bounded Setting (vs. Unbounded Settings)
  15. A Redundancy for Strictly Positive Channels
  16. ...and 7 more sections

Key Result

Lemma 3.1

If a quantum channel $\mathcal{N}: \mathcal{B}(\mathcal{H}_A) \to \mathcal{B}(\mathcal{H}_B)$ is purity-preserving, i.e. it maps any pure state into a pure state, then $\mathcal{N}$ must be either an isometric embedding $\mathcal{N}(\rho) = V \rho V^\dagger$, $V^\dagger V = \mathbbm{1}_A$, or a repl

Theorems & Definitions (53)

  • Lemma 3.1: davies1976quantum
  • Lemma 3.2: BGSW24
  • Theorem 3.3: No Divergence-based Reverse DPI on All States
  • proof
  • Proposition 4.1
  • proof
  • Theorem 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 43 more