Bi-Free Entropy with Respect to Completely Positive Maps

TL;DR

This paper introduces a new concept of bi-free entropy relative to completely positive maps, extending existing free entropy theories and analyzing bi-R-diagonal elements using conjugate variables and Fisher information.

Contribution

It extends non-microstate bi-free entropy and free entropy with respect to completely positive maps, incorporating operator-valued structures and conjugate variables for analytical development.

Findings

Minima of Fisher information at bi-R-diagonal elements

Maxima of bi-free entropy at bi-R-diagonal elements

Extension of entropy and Fisher information concepts to this new setting

Abstract

In this paper, a notion of non-microstate bi-free entropy with respect to completely positive maps is constructed thereby extending the notions of non-microstate bi-free entropy and free entropy with respect to a completely positive map. By extending the operator-valued bi-free structures to allow for more analytical arguments, a notion of conjugate variables is constructed using both moment and cumulant expressions. The notions of free Fisher information and entropy are then extended to this setting and used to show minima of the Fisher information and maxima of the non-microstate bi-free entropy at bi-R-diagonal elements.