Summing free unitary Brownian motions with applications to quantum information
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Motivated by quantum information theory, we introduce a dynamical random state 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 <cit.> 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.
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