Summing free unitary Brownian motions with applications to quantum information

Motivated by quantum information theory, we introduce a dynamical random density matrix built out of the sum of k≥ 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≥ 3 the one derived in Demni et al. (Indiana Univ Math J 61:1351–1368, 2012) 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 satisfied by the moment generating function of this non normal operator and for which we determine the corresponding characteristic curves. As an application of our results, we compute the average purity and the entanglement entropy of the large-size limiting density matrix.