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Trace- and improved data processing inequalities for von Neumann algebras

Stefan Hollands

Abstract

We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The inequality, which generalizes previous work by Sutter et al. on finite dimensional density matrices, yields a bound how well a quantum state can be recovered after it has been passed through a channel. The natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III. Along the way we generalize various multi-trace inequalities to general von Neumann algebras.

Trace- and improved data processing inequalities for von Neumann algebras

Abstract

We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The inequality, which generalizes previous work by Sutter et al. on finite dimensional density matrices, yields a bound how well a quantum state can be recovered after it has been passed through a channel. The natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III. Along the way we generalize various multi-trace inequalities to general von Neumann algebras.

Paper Structure

This paper contains 9 sections, 12 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Von Neumann algebras and modular theory
  3. Interpolation of non-commutative $L_p$ norms
  4. Multi-trace inequalities for von Neumann algebras
  5. Improved DPI and recovery channels
  6. Relative entropy and measured relative entropy
  7. Petz recovery map
  8. Improved DPI
  9. Weighted $L_p$ spaces AM and variational formulae

Key Result

Lemma 1

Let $|G(z)\rangle$ be a $\mathscr{H}$-valued holomorphic function on the strip ${\mathbb S}_{1/2}=\{0<{\rm Re}z<1/2\}$ that is uniformly bounded in the closure, and let $|\psi\rangle \in \mathscr{H}$ a state of a $\sigma$-finite von Neumann algebra $\mathcal{M}$ in standard form acting on $\mathscr{ Then where

Theorems & Definitions (22)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • ...and 12 more