Quantum Mechanics of Stochastic Systems
Yurang, Kuang
TL;DR
The paper develops a comprehensive quantum-mechanical framework for stochastic systems by treating classical probability laws as spectral outcomes of perturbations to the quantum harmonic oscillator. It formalizes an infinite-dimensional Hilbert-space representation in which stochastic states $\ket{\psi_S}=\sum_n\alpha^{(S)}_n\ket{n}$ encode probability distributions, and introduces exact perturbation potentials to realize Poisson, Binomial, Negative Binomial, and Hypergeometric laws. A key innovation is the modular projection $\hat{R}_M$ that maps to finite-dimensional bases and guarantees True Uniform Random Number Generation (TURNG) via exponential convergence to uniform modulo $M$, removing the need for external whitening. Beyond RNGs, the framework enables quantum probability engineering—physically realizing classical distributions through designed Hamiltonians—and provides a dynamical extension with stochastic time evolution, multi-system correlations, and master-equation connections. Collectively, QMSS bridges quantum dynamics, statistical physics, and experimental probability realization, offering a rigorous route from classical randomness to quantum-inspired computation and certified randomness in quantum laboratories.
Abstract
We develop a fundamental framework for the quantum mechanics of stochastic systems (QMSS), showing that classical discrete stochastic processes emerge naturally as perturbations of the quantum harmonic oscillator (QHO). By constructing exact perturbation potentials that transform QHO eigenstates into stochastic representations, we demonstrate that canonical probability distributions, including Binomial, Negative Binomial, and Poisson, arise from specific modifications of the harmonic potential. Each stochastic system is governed by a Count Operator (N), with probabilities determined by squared amplitudes in a Born-rule-like manner. The framework introduces a complete operator algebra for moment generation and information-theoretic analysis, together with modular projection operators (R_M) that enable finite-dimensional approximations supported by rigorous uniform convergence theorems. This mathematical structure underpins True Uniform Random Number Generation (TURNG) [Kuang, Sci. Rep., 2025], eliminating the need for external whitening processes. Beyond randomness generation, the QMSS framework enables quantum probability engineering: the physical realization of classical distributions through designed quantum perturbations. These results demonstrate that stochastic systems are inherently quantum-mechanical in structure, bridging quantum dynamics, statistical physics, and experimental probability realization.