Stochastic-Quantum Correspondence

• Stochastic-Quantum Correspondence is defined as the mapping between classical stochastic processes and quantum mechanics, demonstrating how randomness induces quantization.
• Rigorous methodologies, including the Madelung transformation and Hilbert-space lifting, derive quantum evolution equations equivalent to Schrödinger and Lindblad forms.
• Quantum simulation techniques leverage this correspondence to efficiently replicate stochastic dynamics on quantum computers, offering new tools for quantum research.

The stochastic-quantum correspondence refers to a broad suite of mathematical, physical, and conceptual frameworks establishing precise connections between stochastic processes and quantum phenomena. These connections range from the emergence of quantum mechanics from classical systems subject to random fields, to rigorous mappings between general (possibly non-Markovian) stochastic dynamics and quantum theory, to quantum simulation of stochastic processes and novel foundational interpretations of quantum probability. The following sections systematically organize the main research lines, methodologies, and key results on the stochastic-quantum correspondence as developed in the contemporary literature.

1. Classical Stochastic Processes as the Origin of Quantum Dynamics

A central historical and technical theme is the emergence of quantum dynamics—specifically the Schrödinger equation—from classical systems coupled to stochastic environments. In "Relevance of stochasticity for the emergence of quantization" (Cetto et al., 2020), Cetto et al. analyze a charged particle subjected to an external potential and a stochastic zero-point field (ZPF), with ZPF statistics fixed by the electromagnetic vacuum spectrum. The dynamics is governed by a stochastic Abraham–Lorentz equation,

$m\ddot{x}(t) = -\nabla V(x(t)) + m\tau\,\dddot{x}(t) + eE(t),$

where $E(t)$ is a Gaussian stationary field with spectrum proportional to $\hbar\omega^3$.

In the long-time, Markovian regime, the effective dynamics reduces to a Langevin equation,

$dx(t) = v(x,t)\,dt + \sqrt{2D}\,dW_t,$

leading to a Fokker–Planck description for the probability density $\rho(x,t)$. By fixing the diffusion coefficient $D = \hbar/2m$, consistent with detailed balance between ZPF-induced diffusion and radiation damping, and invoking the Madelung transformation,

$$\psi(x,t) = \sqrt{\rho(x,t)}\,e^{iS(x,t)/\hbar}, \qquad v(x,t) = \frac{1}{m}\nabla S(x,t),$$

the evolution equations become equivalent to the Schrödinger equation,

$$i\hbar\frac{\partial\psi}{\partial t} = \left[-\frac{\hbar^2}{2m}\nabla^2 + V(x)\right]\psi.$$

The single-valuedness of $\psi$ ensures the Bohr–Sommerfeld quantization condition and a discrete spectrum. Analogous stochastic-induced quantumlike behavior is observed in macroscopic hydrodynamic pilot-wave systems. This line of argument demonstrates that coupling to an appropriate stochastic environment is sufficient for the emergence of quantization and key quantum phenomena (Cetto et al., 2020).

2. Structural Correspondence: Lifting Stochastic Processes to Quantum Formalism

The general mathematical foundation for the stochastic-quantum correspondence is established by theorems rigorously relating arbitrary stochastic processes—Markovian or non-Markovian, classical or generalized—to Hilbert-space-based quantum systems.

Barandes introduces in "The Stochastic-Quantum Theorem" (Barandes, 2023, Barandes, 2023) a formalism in which any process on a finite configuration space with transition matrices $P(t)$, where $P(t)$ is column-stochastic, can be factorized entrywise as $P_{ij}(t) = |U_{ij}(t)|^2$ for a (possibly non-unique) complex matrix $U(t)$. Considering the initial distribution $\rho(0)$, the induced density operator evolves as $\rho(t) = U(t)\rho(0)U(t)^\dagger$, and measurement probabilities are recovered via the Born rule,

$p_i(t) = \operatorname{Tr}[P_i \rho(t)],$

where $P_i$ are diagonal projectors. For general, possibly non-Markovian, stochastic systems, a Stinespring dilation shows that the process is always the marginal of unitary evolution on a larger Hilbert space. This framework encompasses both classical Markov chains and more general stochastic evolutions, and directly yields quantum dynamics (Schrödinger or Lindblad form) in the unistochastic (unitarity-preserving) case. It also implies that any classical stochastic process can be simulated exactly by a quantum computer, with only a polynomial overhead in Hilbert space dimension (Barandes, 2023, Barandes, 2023).

3. Non-Markovian "Indivisible" Stochastic Processes and Quantum Nonclassicality

Quantum processes famously violate Markov assumptions: the divisibility condition for transition probabilities fails. This is formalized by the notion of "indivisible" stochastic processes (Barandes, 27 Jul 2025). Here, the evolution of the system is specified by transition probabilities $p(i,t|j,t_0)$, but with no requirement of factorization through intermediate times. The law of total probability holds,

$p(i,t) = \sum_j p(i,t|j,t_0)p(j,t_0),$

but Chapman–Kolmogorov composition can fail. The Hilbert-space lifting constructed in Section 2 remains valid, and quantum phenomena like interference, entanglement, and collapse can be interpreted as natural features of such indivisible stochastic processes. The "indivisibility" is operationally testable: violation of Kolmogorov conditions by the induced dynamical map witnesses the genuinely quantum character of the evolution (Chruściński et al., 2013, Barandes, 27 Jul 2025).

Gauge invariances, such as the Schur–Hadamard entrywise phase and ordinary Foldy–Wouthuysen unitaries, emerge as structural redundancies in the stochastic–quantum lifting, with all observable probabilities remaining invariant.

4. Stochastic Calculus, Path Integrals, and Quantum Field Theory Correspondence

Several approaches render quantum amplitudes as generalized functionals or "square roots" of classical stochastic processes. In Frasca's construction (Frasca, 2012), quantum mechanics is obtained by taking the square root (in a generalized Itō sense) of a real Wiener process, requiring the introduction of complex-valued stochastic calculus and a hidden Bernoulli (coin-toss) process to consistently define amplitudes. The resulting forward equation for the complex probability amplitude—after suitable identification of coefficients—maps directly to the Schrödinger equation.

Algebraic quantum field theory techniques have rigorously established the correspondence between the solution theory for stochastic differential equations and the functional integral (MSR/Janssen–de Dominicis) formulation of statistical field theories (Bonicelli et al., 2023). For a broad class of (nonlinear) SDEs, all n-point correlators in the SDE and MSR formulations agree order by order in perturbation theory.

Stochastic calculus constructions extend to relativistic field theory: general Itō-diffusion (complex, forward–backward) processes yield the fundamental quantum wave equations (Schrödinger and Klein–Gordon) via the stochastic Hamilton–Jacobi principle, with careful handling of circulation quantization to enforce single-valuedness (Kuipers, 2023).

5. Stochastic Interpretations of Quantum Probability and Dynamics

Recent stochastic approaches reinterpret the wave function as a realization of suitable underlying stochastic processes. Several models emerge:

• Norm-preserving phase-noise SDEs: Treating the wave function as a complex-valued stochastic variable whose norm is preserved under pure phase randomization. The covariance matrix of this process evolves by the quantum Liouville-von Neumann equation, and ensemble averages reproduce Born-rule probabilities (Oliveira, 7 Oct 2025, Oliveira, 2023, Olliveira, 2023).
• Classical phase-space representations: Systems are constructed by canonical mapping into complex variables, and quantum expectations are recovered as averages or moments of classical observables under Gaussian distributions induced by underlying stochastic processes (Oliveira, 2023, Olliveira, 2023).
• Bilinear two-process representations: The density matrix is built from two independent Markov jump processes, which never individually superpose, but whose bilinear correlations reproduce entanglement and interference. This "bra-ket" interpretation provides an alternative ontological perspective with a strictly probabilistic underpinning (Öttinger, 2023).
• Dynamic programming and HJB: By a phase-density transform, the quantum Hamilton–Jacobi system is recast as a stochastic Hamilton–Jacobi–Bellman equation, elimination of the quantum potential in the recursion, and a retrocausal, local interpretation of quantum mechanics at the ensemble level (Brownstein, 8 Jan 2024).

6. Quantum Simulation of Stochastic Processes and Quantum Algorithms

The stochastic–quantum correspondence enables efficient simulation and engineering of stochastic processes using quantum systems.

• Quantum mechanics of stochastic systems (QMSS): By perturbing the quantum harmonic oscillator, exact ground states with amplitudes squared to classical probability distributions (binomial, negative binomial, Poisson) are constructed, governed by a count operator. Modular projections yield uniform random number generation without external whitening, with rigorous uniformity guarantees (Yurang et al., 25 Oct 2025).
• Open quantum systems and SDEs: For bosonic systems with environmental coupling, the Lindblad master equation is mapped exactly to a finite-dimensional Itô SDE. Mean-field limits recover deterministic PDEs. Quantum algorithms simulate nonlinear SDEs at polynomial cost in the logarithm of the state-space size, exploiting high-dimensional quantum encoding (Engel et al., 2023).
• Quantum trajectories and memory-efficient stochastic simulators: Embedding hidden semi-Markov models as quantum jump trajectories—emulated via continuous-time Lindblad equations—yields quantum simulators with lower memory overhead and complexity than their classical $\varepsilon$-machine counterparts, particularly for processes with diverging classical complexity (Elliott et al., 7 Feb 2024).
• Stochastic classical emulators of quantum algorithms: Quantum states and universal gates can be represented as higher-order derivatives of classical stochastic distributions ("grabits"), enabling simulation of quantum circuits through stochastic maps. This approach reproduces quantum dynamics with explicit Hilbert-space structure and elucidates the resource cost of destructive interference (Braun et al., 2021, Chruściński et al., 2013).

7. Extensions, Limitations, and Open Questions

While the stochastic–quantum correspondence provides powerful unifying frameworks, substantial open problems and limitations remain:

• For fully relativistic and field-theoretical systems, rigorous stochastic quantization remains challenging, especially regarding spin, gauge symmetry, and interacting quantum field theories (Cetto et al., 2020, Kuipers, 2023).
• The domain of validity of stochastic representations depends critically on the preservation of key constraints, such as norm-preservation, proper closure of moment hierarchies, and statistical consistency.
• For non-quadratic Hamiltonians, exact mappings may fail, but explicit small parameters enable controlled approximation schemes (e.g., in quantum Brownian motion with strong bath-induced quasidiagonality) (Kondaurov et al., 9 Dec 2025).
• The operational distinction between quantum and classical stochastic evolutions can be sharply drawn in finite-level systems by explicit violation of Kolmogorov stochasticity in the quantum-induced dynamical maps (Chruściński et al., 2013).
• The practical realization of quantum-inspired uniform random number generators based on modular projections raises new engineering and foundational questions in both physics and information theory (Yurang et al., 25 Oct 2025).

The stochastic–quantum correspondence is a unifying principle animating modern research into the foundations, simulation, and engineering of quantum systems, as well as a touchstone for debates concerning the nature of quantum indeterminacy, the emergence of quantization, and the deep algebraic structures underlying quantum theory. The connection between indivisible (non-Markovian) stochastic laws and quantum theoretical structure is now recognized as an essential perspective in both conceptual and applied quantum science.