Analogues of Entropy in Bi-Free Probability Theory: Non-Microstate

Abstract

In this paper, we extend the notion of non-microstate free entropy to the bi-free setting. Using a diagrammatic approach involving bi-non-crossing diagrams, bi-free difference quotients are constructed as analogues of the free partial derivations. Adjoints of bi-free difference quotients are discussed and used to define bi-free conjugate variables. Notions of bi-free Fisher information, non-microstate bi-free entropy, and non-microstate bi-free entropy dimension are defined and known properties in the free setting are extended to the bi-free setting.