QLunch with Cambyse Rouzé

Speaker: Cambyse Rouzé, Technische Universität München

Title: Approximate tensorization of the relative entropy for noncommuting conditional expectations

Abstract:

In the last few decades, the relative entropy has proven to be a fundamental object in various fields of mathematics and theoretical physics. Its quantum analogue characterizes the optimal rate at which two different states of a system can be discriminated when an arbitrary number of identical copies is available. In this talk, I will derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. The latter inequality, usually referred to as approximate tensorization (AT) of the relative entropy, can be expressed as a lower bound on the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive error terms. In absence of an additive error, AT was proved to be the key step in modern proofs of the entropic exponential decay of dissipative processes in lattice spin systems in the one phase region towards their equilibrium. Entropic uncertainty relations provide another application of our inequality, where we get tightenings of previous known bounds near the completely mixed state. This talk is based on the following preprint: arXiv:2001.07981v1.