Seminar talk by Tiju Cherian John, University of Arizona on 20th July at 14:00 hrs

Speaker: Tiju Cherian John
Affiliation: Wyant College of Optical Sciences
The University of Arizona
Title: A quantum = classical result on $f$-divergences and some applications
Venue: TBA
Date and Time: 20th July at 14:00 hrs

Abstract

Abstract: In classical information theory, $f$-divergence of two probability distributions is a parent quantity that generates several useful relative entropic quantities. An analogous quantity called quantum $f$-divergence of two quantum states is being studied in quantum information theory. We prove that the quantum $f$-divergence of two states is equal to the classical $f$-divergence of the corresponding Nussbaum-Szko{\l}a distributions. This provides a general framework for studying certain properties of quantum entropic quantities using the corresponding classical entities. We show three important applications of the main result:

1. Several quantum $f$-divergence inequalities are obtained from their classical counterparts,
2. New formulas for Petz-R\'enyi and Umegaki relative entropy are derived by connecting the main result to the idea of distribution of a positive selfadjoint operator with respect to a state,
3. Precise range of the values of the parameter $\alpha$ such that Petz-Rényi $\alpha$-relative entropy $D_{\alpha}(\rho||\sigma)$ of two faithful displaced thermal states is finite, are obtained. This result is particularly useful in the light of recent operational interpretation of Petz-Rényi $\alpha$-relative entropy in connection with certain optimal quantum encoding (Sci. Rep. 7, 14765 (2017)). Along the way, we prove a special case of a conjecture of Seshadreesan, Lami and Wilde (J. Math. Phys. 59, 072204 (2018)).

This is a joint work with George Androulakis.