Free Mutual Information for Two Projections

Abstract

The present paper provides a proof of i * (CP + C(I − P ); CQ + C(I − Q)) = −χ 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.

Schur-Weyl duality and the Product of randomly-rotated symmetries by a unitary Brownian motion

“…Nonetheless, when each subalgebra is generated by a single orthogonal projection, one gets an instance of the so-called Voiculescu's liberation process ( [27]) or in a different guise, the free Jacobi process ( [9]). In this respect, an extensive spectral study of fixed-time marginals of this process was performed in [16,17,12,13,4,18,10,11,23,24,25]. In particular, it turns out that the spectral dynamics of the free Jacobi process are governed by that of A t := RU t SU ⋆ t , t ≥ 0, where R, S are self-adjoint symmetries which are freely-independent from {U, U ⋆ }.…”