Free mutual information for two projections

TL;DR

This paper proves a formula relating free mutual information and orbital free entropy for two projections, using an analytic approach based on subordination in free probability theory.

Contribution

It provides a rigorous proof of the equality between free mutual information and orbital free entropy for two projections without additional assumptions.

Findings

Established the equality $i^*( ext{C}P+ ext{C}(I-P); ext{C}Q+ ext{C}(I-Q) )=- ext{χ}_{orb}(P,Q)$.

Developed an analytic proof using subordination for the liberation process of symmetries.

Extended the understanding of free mutual information in the context of projections.

Abstract

The present paper provides a proof of $i^*( \mathbb{C}P+\mathbb{C}(I-P); \mathbb{C}Q+\mathbb{C}(I-Q) )=-\chi_{orb}(P,Q)$ for two projections $P,Q$ without any extra assumptions. An analytic approach is adopted to the proof, based on a subordination result for the liberation process of symmetries associated with $P,Q$.