The stochastic-quantum correspondence rewrites any quantum unitary as a matrix of squared entries, making quantum systems formally equivalent to indivisible stochastic processes, but the equivalence is definitional and the unitarity claim overreaches.
Classical stochastic representation of quantum mechanics
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We show that the dynamics of a quantum system can be represented by the dynamics of an underlying classical systems obeying the Hamilton equations of motion. This is achieved by transforming the phase space of dimension $2n$ into a Hilbert space of dimension $n$ which is obtained by a peculiar canonical transformation that changes a pair of real canonical variables into a pair of complex canonical variables which are complex conjugate of each other. The probabilistic character of quantum mechanics is devised by treating the wave function as a stochastic variable. The dynamics of the underlying system is chosen so as to preserve the norm of the state vector.
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Quantum Systems as Indivisible Stochastic Processes
The stochastic-quantum correspondence rewrites any quantum unitary as a matrix of squared entries, making quantum systems formally equivalent to indivisible stochastic processes, but the equivalence is definitional and the unitarity claim overreaches.