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

## TL;DR

This paper extends the concept of non-microstate free entropy to the bi-free setting using diagrammatic methods, defining bi-free conjugate variables, Fisher information, and entropy, thereby broadening the theoretical framework of free probability.

## Contribution

It introduces a bi-free analogue of non-microstate free entropy, including bi-free difference quotients, conjugate variables, and Fisher information, expanding free entropy theory.

## Key findings

- Defined bi-free conjugate variables.
- Extended properties of free entropy to bi-free setting.
- Constructed bi-free difference quotients and Fisher information.

## 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 and non-microstate entropy are defined and properties of free entropy are extended to the bi-free setting.