For finite reversible Markov chains, the stochastic semigroup and a unitary quantum evolution are real and imaginary slices of one complex-orthogonal group W_z = e^{zK}; for the Ehrenfest urn the classical law is the Born law under the clock θ(t) = arccos(e^{-2t}).
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The pseudo-quantum representation of finite reversible Markov chains
For finite reversible Markov chains, the stochastic semigroup and a unitary quantum evolution are real and imaginary slices of one complex-orthogonal group W_z = e^{zK}; for the Ehrenfest urn the classical law is the Born law under the clock θ(t) = arccos(e^{-2t}).