Quantum Brascamp-Lieb Dualities

TL;DR

This paper introduces a quantum duality framework linking quantum entropy inequalities to matrix exponential inequalities, with applications in quantum information theory including new uncertainty relations for Gaussian quantum operations.

Contribution

It develops a fully quantum version of Brascamp-Lieb duality, connecting entropy inequalities to matrix exponential inequalities, and demonstrates applications in quantum information theory.

Findings

New quantum duality between entropy and matrix exponential inequalities

Derivation of novel uncertainty relations for Gaussian quantum operations

Examples include entropic uncertainty and data-processing inequalities

Abstract

Brascamp-Lieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theory -- including entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of `geometric' Brascamp-Lieb inequalities.