Remarks on free mutual information and orbital free entropy

TL;DR

This paper proves a key relationship between free mutual information and orbital free entropy for pairs of projections with equal trace, introducing new observations and extending to the general case.

Contribution

It provides a proof of the equality between free mutual information and orbital free entropy for certain projections, including new subordination results and general case analysis.

Findings

Established the equality $i^* = - ext{orbital free entropy}$ for projections with trace 1/2

Introduced a subordination result using Loewner equations

Extended analysis to the general case of projections

Abstract

The present notes provide a proof of $i^*(\mathbb{C}P+\mathbb{C}(I-P)\,;\mathbb{C}Q+\mathbb{C}(I-Q)) = -\chi_\mathrm{orb}(P,Q)$ for any pair of projections $P,Q$ with $\tau(P)=\tau(Q)=1/2$. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.