Quantum Brascamp–Lieb Dualities

Communications in Mathematical Physics, vol. 401, pp. 1807–1830

Abstract

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.

Authors 3

1. Mario Berta corresponding Aachen Institute for Quantum Information

   RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information, RWTH Aachen University, Aachen, Germany

2. David F. Sutter corresponding

   IBM Research - Zurich

   Affiliation as printed

   IBM Quantum, IBM Research Europe, Zurich, Switzerland

3. Michael Walter

   University of Amsterdam · Ruhr University Bochum · Institute for Theoretical Physics Amsterdam · QuSoft

   Affiliation as printed

   Faculty of Computer Science, Ruhr University Bochum, Bochum, Germany

   Korteweg-de Vries Institute for Mathematics, Institute for Theoretical Physics, Institute for Language, Logic and Computation, and QuSoft, University of Amsterdam, Amsterdam, The Netherlands