# Liberation, free mutual information and orbital free entropy

## TL;DR

This paper explores the connections between liberation processes for projections and symmetries in free probability, relating spectral measures, deriving PDEs for unitary processes, and providing an improved proof of an entropy-related formula.

## Contribution

It introduces new relationships between spectral measures of liberation processes, derives PDEs for Herglotz transforms, and offers an improved proof of an entropy formula in free probability.

## Key findings

- Relations between spectral measures $\mu_t$ and $
u_t$ established.
- Derived PDE for the Herglotz transform of unitary processes.
- Improved proof of the entropy formula involving projections and symmetries.

## Abstract

We present here some connections between the liberation process for projections $(P,Q)\mapsto(P,U_tQU_t^*)$ and its counterpart $(R,S)\mapsto(R,U_tSU_t^*)$ for symmetries when the projections $\{P,Q\}$ and the symmetries $\{R,S\}$ are associated, where $U_t$ is a free unitary Brownian motion freely independent from $\{P,Q\}$ (and so $\{R,S\}$). We relate the moments of their actions on the operators $X_t:=PU_tQU_t^*$ and $Y_t:=U_tRU_t^*S$ and use this to prove a relationship between the corresponding spectral measures (hereafter $\mu_t$ and $\nu_t$). On the other hand, we focus in the process of unitary random variables $Y_t$ in the case of arbitrary trace values $\tau(R),\tau(S)$. More precisely, we use stochastic calculus to derive a partial differential equation (PDE for short) for its Herglotz transform and use it to develop subordination results in terms of L\"owner equations. The paper is closed with an improved proof of $i^*\left( \mathbb{C}P+\mathbb{C}(I-P); \mathbb{C}Q+\mathbb{C}(I-Q) \right)=-\chi_{orb}\left(P,Q\right)$ as an application.