Quantum f-Divergences in von Neumann Algebras: Reversibility of Quantum Operations

Description

Preface Contents 1 Introduction 2 Standard f-Divergences 2.1 Definition 2.2 Variational Expression of Standard f-Divergences 2.3 Properties of Standard f-Divergences 3 Rényi Divergences and Sandwiched Rényi Divergences 3.1 Rényi Divergences 3.2 Description of Rényi Divergences 3.3 Sandwiched Rényi Divergences 4 Maximal f-Divergences 4.1 Definition and Basic Properties 4.2 Further Properties of Maximal f-Divergences 4.3 Minimal Reverse Test 5 Measured f-Divergences 5.1 Definition 5.2 Variational Expressions 5.3 Optimal Measurements 6 Reversibility and Quantum Divergences 6.1 Petz' Recovery Map 6.2 A Technical Lemma 6.3 Preservation of Connes' Cocycle Derivatives 6.4 Reversibility via Standard f-Divergences 6.5 Reversibility via Sandwiched Rényi Divergences 7 Reversibility and Measurements 7.1 Approximation of Connes' Cocycle Derivatives and Approximate Reversibility 7.2 Reversibility via Measurements 8 Preservation of Maximal f-Divergences 8.1 Maximal f-Divergences and Operator Connections 8.2 Proof of the Theorem A Preliminaries on von Neumann Algebras A.1 Introduction of von Neumann Algebras A.2 Tomita–Takesaki Modular Theory A.3 Standard Forms A.4 Relative Modular Operators A.5 τ-Measurable Operators A.6 Haagerup's Lp-Spaces A.7 Connes' Cocycle Derivatives A.8 Kosaki's Lp-Spaces B Preliminaries on Positive Self-Adjoint Operators C Operator Convex Functions on (0,1) D Operator Connections of Normal Positive Functionals References Index