Operator-valued Schatten spaces and quantum entropies

TL;DR

This paper introduces operator-valued Schatten spaces, explains their theoretical framework, and demonstrates their application in quantum information theory, including a new continuity bound for quantum conditional Rényi entropy.

Contribution

It provides a self-contained exposition of Pisier's operator-valued Schatten spaces and applies this theory to derive new results in quantum information theory.

Findings

Established a new uniform continuity bound for quantum conditional Rényi entropy

Provided an accessible exposition of Pisier's theory for the QIT community

Demonstrated applications of operator-valued Schatten spaces in quantum entropies

Abstract

Operator-valued Schatten spaces were introduced by G. Pisier as a noncommutative counterpart of vector-valued $\ell_p$-spaces. This family of operator spaces forms an interpolation scale which makes it a powerful and convenient tool in a variety of applications. In particular, as the norms coming from this family naturally appear in the definition of certain entropic quantities in Quantum Information Theory (QIT), one may apply Pisier's theory to establish some features of those quantities. Nevertheless, it could be quite challenging to follow the proofs of the main results of this theory from the existing literature. In this article, we attempt to fill this gap by presenting the underlying concepts and ideas of Pisier's theory in a self-contained way which we hope to be more accessible, especially for the QIT community at large. Furthermore, we describe some applications of this theory in QIT. In particular, we prove a new uniform continuity bound for the quantum conditional R\'enyi entropy.