Summing free unitary Brownian motions with applications to quantum information

TL;DR

This paper introduces a new dynamical random state based on sums of independent unitary Brownian motions, explores its spectral distribution in the large size limit, and develops analytical tools like PDEs for associated moment generating functions.

Contribution

It extends the understanding of free Jacobi processes and introduces a binomial expansion for moments in the context of sums of multiple free unitary Brownian motions.

Findings

Spectral distribution matches free Jacobi process in large size limit.

Derived a binomial-type expansion for moments of the free Jacobi process for any k ≥ 3.

Established PDEs for the moment generating function of a non-normal operator from splitting a projection.

Abstract

Motivated by quantum information theory, we introduce a dynamical random state built out of the sum of $k \geq 2$ independent unitary Brownian motions. In the large size limit, its spectral distribution equals, up to a normalising factor, that of the free Jacobi process associated with a single self-adjoint projection with trace $1/k$. Using free stochastic calculus, we extend this equality to the radial part of the free average of $k$ free unitary Brownian motions and to the free Jacobi process associated with two self-adjoint projections with trace $1/k$, provided the initial distributions coincide. In the single projection case, we derive a binomial-type expansion of the moments of the free Jacobi process which extends to any $k \geq 3$ the one derived in \cite {DHH} in the special case $k=2$. Doing so give rise to a non normal (except for $k=2$) operator arising from the splitting of a self-adjoint projection into the convex sum of $k$ unitary operators. This binomial expansion is then used to derive a pde for the moment generating function of this non normal operator and for which we determine the corresponding characteristic curves.