# Analogues of Entropy in Bi-Free Probability Theory: Microstates

## TL;DR

This paper extends the concept of microstate free entropy to bi-free probability, developing properties, calculations for bi-free central limit distributions, and an orbital version to better understand bi-freeness.

## Contribution

It introduces the microstate bi-free entropy, explores its properties, and provides new tools like the orbital version for analyzing bi-freeness.

## Key findings

- Microstate bi-free entropy is additive under certain conditions.
- Explicit computation of bi-free entropy for central limit distributions.
- Orbital bi-free entropy offers a tighter subadditivity bound.

## Abstract

In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are developed, such as the behaviour of the entropy when transformations on the left variables or on the right variables are performed. In addition, the microstate bi-free entropy is demonstrated to be additive over bi-free collections provided additional regularity assumptions are included and is computed for all bi-free central limit distributions. Moreover, an orbital version of bi-free entropy is examined which provides a tighter upper bound for the subadditivity of microstate bi-free entropy and provides an alternate characterization of bi-freeness in certain settings.