Bayes’ rule connects forward and reverse processes in classical probability theory, and its quantum analogue has been discussed in terms of the Petz (transpose) map. For quantum dynamics governed by the Lindblad equation, the corresponding Petz map can also be written in Lindblad form. In classical stochastic systems, the analogue of the Lindblad equation is the Fokker–Planck equation, and applying Bayes’ rule to it yields the reverse diffusion equation underlying modern diffusion-based generative models. It is known that a semiclassical approximation of the Lindblad equation yields the Fokker–Planck equation for the Wigner function, which is a quasiprobability distribution defined on phase space as the Wigner transform of the density operator. Here we demonstrate that applying the same approximation to the Lindblad equation associated with the Petz map produces an equation that coincides with that obtained from the Fokker–Planck equation via Bayes’ rule. This finding establishes a direct correspondence between the Petz map and Bayes’ rule, unifying quantum reversibility with classical reverse diffusion.
Bayes’ rule connects forward and reverse processes in classical probability theory, and its quantum analogue has been discussed in terms of the Petz (transpose) map. For quantum dynamics governed by the Lindblad equation, the corresponding Petz map can also be written in Lindblad form. In classical stochastic systems, the analogue of the Lindblad equation is the Fokker–Planck equation, and applying Bayes’ rule to it yields the reverse diffusion equation underlying modern diffusion-based generative models. It is known that a semiclassical approximation of the Lindblad equation yields the Fokker–Planck equation for the Wigner function, which is a quasiprobability distribution defined on phase space as the Wigner transform of the density operator. Here we demonstrate that applying the same approximation to the Lindblad equation associated with the Petz map produces an equation that coincides with that obtained from the Fokker–Planck equation via Bayes’ rule. This finding establishes a direct correspondence between the Petz map and Bayes’ rule, unifying quantum reversibility with classical reverse diffusion.
Bayes' rule is central to classical statistical inference: given a prior and a stochastic model, it determines the corresponding retrodictive conditional probabilities. In quantum theory, however, the question of whether an equally general rule of retrodiction exists and, if so, whether it is uniquely determined remains nontrivial. Here we prove that a retrodiction assignment defined for all priors and channels, which takes each predicted output state back to its prior, must coincide with the Petz transpose map if it is involutive, respects sequential and parallel composition, and agrees with unitary inversion. The theorem holds for channels between finite-dimensional full matrix algebras, with the input and output spaces restricted to the supports of the corresponding reference states. In particular, affinity under probabilistic mixtures and the characteristic square-root dependence on the reference states follow from these requirements rather than being postulated. Without involutivity, the remaining requirements characterize a family of doubly rotated Petz maps. Our result thus explains why the Petz transpose map arises in seemingly different areas of quantum theory: it is the unique quantum analogue of Bayes' rule compatible with these consistency requirements.
Reversibility lies at the heart of physics and information theory, linking entropy, efficiency and information recovery. Yet, definitions of a “reverse process” have been context-specific, relying on physical assumptions rather than a general framework. This thesis develops a universal recipe for reverse processes using Bayesian inference, formalized through Bayes’ rule and the Petz recovery map. We first show how these tools recover conventional notions of reversal while yielding new insights, and justify the Petz transpose as the quantum analog of Bayes’ rule via quasiprobability representations. We then compare Bayesian reversibility with dilation-based approaches, proving their equivalence through the role of correlations. This motivates the notion of tabletop time-reversible processes, special cases where correlations can be neglected with respect to reversal. Beyond the core results, we apply these methods to fluctuation relations, quantum thermal operations, and the Loschmidt paradox, and reflect on implications for the philosophy of time. In sum, Bayesian inference provides a regime-independent foundation for defining and analyzing (ir)reversibility across physics and information theory.
We develop a Bayesian formulation of diffusive quantum-classical dynamics by treating the wave function and classical variables as components of an ordinary stochastic process. The joint probability density P(psi,x,t) obeys a classical Fokker-Planck equation, while the quantum state appears as its second moment. Requiring autonomous second-moment evolution and trace preservation, together with a full-column-rank quantum-classical noise correlation matrix, forces the ensemble dynamics into hybrid Lindblad form. The same result also determines the general class of diffusive stochastic unravelings compatible with this hybrid evolution. Notably, this construction makes positivity immediate and does not take complete positivity or an underlying unitary quantum dilation as assumptions, but instead relies on the presence of a sufficiently coupled classical sector.
We formulate quantum measurement, filtering and smoothing of hybrid quantum-classical dynamics as standard Bayesian inference and show how to correlation between the quantum and classical noise fully explains the differences between quantum smoothing and smoothing in classical time series models. We recover the stochastic master equation from the Kushner-Stratonovich equation with correlated noise and show how the quantum effect operator is related to the Bayesian backward message through the adjoint dynamics of the linear unraveling. The Bayesian posterior also defines a smoothed density matrix and, more generally, a posterior distribution over latent quantum-classical trajectories.
These quantities can be approximated with standard particle filtering and smoothing methods. Numerical examples show that smoothing improves reconstruction of a hidden quantum-classical trajectory and that the full trajectory posterior can retain structure, such as multimodality, that is absent from its density-matrix second moment.
Bayes' rule plays a crucial role in logical inference in information and physical sciences alike.Its extension into the quantum regime has been the subject of several recent works.These quantum versions of Bayes' rule have been expressed in the language of Hilbert spaces.In this paper, we derive the expression for the Petz recovery map within any quasiprobability representation, with explicit formulas for the two canonical choices of "normal quasiprobability representations" (which include discrete-Wigner representations) and of representations based on symmetric informationally complete positive operator-valued measures (SIC-POVMs).By using the same mathematical syntax of (quasi)stochastic matrices acting on (quasi)stochastic vectors, the core difference in logical inference between classical and quantum theory is found in the manipulation of the reference prior rather than in the representation of the channel.
We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
A quantum state can be written in phase space, but the resulting object is not generally the probability density of a positive stochastic process on ordinary phase space. We spell this out for Wigner dynamics. If a positive phase-space process is required only to reproduce the Born density after integrating over momentum, the requirement fixes only an integrated current; the local drift and diffusion remain underdetermined. If one instead requires all Weyl-ordered expectation values, the phase-space object is fixed to be the Wigner function. For non-quadratic potentials the Wigner--Moyal generator contains higher-order, signed momentum-transfer terms, so it is not the Fokker--Planck generator of a positive Brownian diffusion. The exact Wigner function must therefore be reconstructed, in a stochastic representation, as a weighted empirical measure \[ \FW(\z,t)=\E_{\Pp}[W_tδ(\z-\z_t)], \qquad \z=(q,p), \] rather than the unweighted density of sampled carrier trajectories. With classical Hamiltonian flow as the carrier, all nonclassical correction beyond classical transport sits in the Moyal residual and can be represented by signed weights or branching events. The same split defines a residual diagnostic that vanishes for quadratic Hamiltonians and measures what classical carrier transport misses in anharmonic dynamics. The formulation also gives a forward--reverse relation for signed Wigner path measures. The ratio of forward and reversed contributions separates into a positive magnitude factor and a sign factor. This sign records the parity of the Wigner interference contribution; it is not a thermodynamic entropy production.