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.
The Importance of Being Unistochastic
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A bistochastic matrix is a square matrix with positive entries such that rows and columns sum to unity. A unistochastic matrix is a bistochastic matrix whose matrix elements are the absolute values squared of a unitary matrix. We can now ask questions such as when a given bistochastic matrix is unistochastic. I review these questions: Why they are asked, why they are difficult to answer, and what is known about them.
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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.