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Approximate recovery and relative entropy I. general von Neumann subalgebras

Thomas Faulkner, Stefan Hollands, Brian Swingle, Yixu Wang

Abstract

We prove the existence of a universal recovery channel that approximately recovers states on a v. Neumann subalgebra when the change in relative entropy, with respect to a fixed reference state, is small. Our result is a generalization of previous results that applied to type-I v. Neumann algebras by Junge at al. [arXiv:1509.07127]. We broadly follow their proof strategy but consider here arbitrary v. Neumann algebras, where qualitatively new issues arise. Our results hinge on the construction of certain analytic vectors and computations/estimations of their Araki-Masuda $L_p$ norms. We comment on applications to the quantum null energy condition.

Approximate recovery and relative entropy I. general von Neumann subalgebras

Abstract

We prove the existence of a universal recovery channel that approximately recovers states on a v. Neumann subalgebra when the change in relative entropy, with respect to a fixed reference state, is small. Our result is a generalization of previous results that applied to type-I v. Neumann algebras by Junge at al. [arXiv:1509.07127]. We broadly follow their proof strategy but consider here arbitrary v. Neumann algebras, where qualitatively new issues arise. Our results hinge on the construction of certain analytic vectors and computations/estimations of their Araki-Masuda norms. We comment on applications to the quantum null energy condition.

Paper Structure

This paper contains 26 sections, 23 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and main results
  3. Tomita-Takesaki theory
  4. Inclusions of v. Neumann algebras and Petz map
  5. Main theorems
  6. Proof of main theorems
  7. Isometries $V_\psi$ for general states, notation
  8. $L_p$ spaces, fidelity and relative entropy
  9. Exact recoverability/sufficiency
  10. Interpolating vector
  11. Strengthened monotonicity
  12. Basic strategy
  13. Filtering and continuity
  14. Updated interpolating vector
  15. $L_p$ norms of updated interpolating vector
  16. ...and 11 more sections

Key Result

Theorem 1

Montonicity of relative entropy can be strengthened to where we assume that $\rho,\sigma$ are normal, $\sigma$ is faithful and where $\alpha_\sigma^t : \mathcal{A} \rightarrow \mathcal{B}$ is the rotated Petz map, defined as $p(t)$ is the normalized probability density defined by $\varsigma^{\sigma, {\mathcal{A}}}_t$ resp. $\varsigma^{\sigma, \mathcal{B}}_t$ are the modular flows of $\sigma$ on

Theorems & Definitions (53)

  • Definition 1
  • Theorem 1: Faithful case
  • Lemma 1
  • proof
  • Corollary 1: Theorem 1 in the non-faithful case
  • Theorem 2
  • Remark 1
  • Lemma 2
  • proof
  • Definition 2
  • ...and 43 more