Every finite stochastic matrix is a marginal of a larger unistochastic matrix, so classical Markov transitions can be represented by unitary quantum evolution with a fixed ancilla.
Random unistochastic matrices
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abstract
An ensemble of random unistochastic (orthostochastic) matrices is defined by taking squared moduli of elements of random unitary (orthogonal) matrices distributed according to the Haar measure on U(N) (or O(N), respectively). An ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble. We study the distribution of complex eigenvalues of bistochastic, unistochastic and ortostochastic matrices in the complex plane. We compute averages (entropy, traces) over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices.
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The Born Representation Theorem and the Unistochastic Theorem
Every finite stochastic matrix is a marginal of a larger unistochastic matrix, so classical Markov transitions can be represented by unitary quantum evolution with a fixed ancilla.