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arXiv:2608.16941

A formal correspondence between Bayesian inference problems and the Heisenberg representation

C. Abugattas Chacoff

quant-phphysics.data-anSubmitted Aug 11, 2026• Announced Aug 13, 2026• Revised Sep 15, 2026 (v2)
v1 · Aug 11, 2026v2 · Sep 15, 2026
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Abstract

The proposed formulation establishes a formal correspondence between the Heisenberg representation and the Bayesian formulation of inverse problems. The likelihood function is first expressed in Hilbert space through a quadratic norm of the difference between the noisy observations and the response of a forward model depending on the unknown parameters, weighted by the inverse of the observational noise covariance. Based on the common mathematical structure of this quadratic form and the observable operator in the Heisenberg representation, the inverse observational noise covariance operator is identified with the initial observable. To establish this correspondence, a stochastic field with Matérn covariance is considered and represented through the Karhunen-Loève expansion. The eigenvalues and eigenfunctions of this expansion are then used to define the state $|Ψ\rangle$ within the proposed formalism. The main results are summarized by four theorems: the unitarity of the evolution operator, the self-adjointness of the Hamiltonian, the conservation of the trace of the observable, and the formal correspondence between the Bayesian likelihood and the Heisenberg observable structure. Limiting cases of the Matérn covariance are analyzed. Finally, numerical examples illustrate the analytical results.

Comments
Revised version 2: corrections to the mathematical formulation and proofs, with improvements in notation and presentation

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The proposed formulation establishes a formal correspondence between the Heisenberg representation and the Bayesian formulation of inverse problems. The likelihood function is first expressed in Hilbert space through a quadratic norm of the difference between the noisy observations and the response of a forward model depending on the unknown parameters, weighted by the inverse of the observational noise covariance. Based on the common mathematical structure of this quadratic form and the observable operator in the Heisenberg representation, the inverse observational noise covariance operator is identified with the initial observable. To establish this correspondence, a stochastic field with Matérn covariance is considered and represented through the Karhunen-Loève expansion. The eigenvalues and eigenfunctions of this expansion are then used to define the state ∣Ψ⟩|Ψ\rangle∣Ψ⟩ within the proposed formalism. The main results are summarized by four theorems: the unitarity of the evolution operator, the self-adjointness of the Hamiltonian, the conservation of the trace of the observable, and the formal correspondence between the Bayesian likelihood and the Heisenberg observable structure. Limiting cases of the Matérn covariance are analyzed. Finally, numerical examples illustrate the analytical results.
The definition and some general properties of the generalized Weyl correspondence between stochastic processes and operator-valued real functions on a Hilbert space plus a trace class operator are given. The relation between the derivatives of an operator-valued function and the derivative of the corresponding stochastic process are studied. When the operator-valued function is the position (or momentum) in the Heisenberg picture, a condition for the positivity of the joint distribution functions of the corresponding process is given, provided that the evolution Hamiltonian be quadratic in the position and momentum. Finally, the case of an arbitrary Hamiltonian evolution for the position operator is studied and the two-dimensional density functions of the process is related to the Wigner function associated to some state ρ̂, and a necessary condition for the positivity of the densities is given.
This paper extends the Karhunen-Loeve representation from classical Gaussian random processes to quantum Wiener processes which model external bosonic fields for open quantum systems. The resulting expansion of the quantum Wiener process in the vacuum state is organised as a series of sinusoidal functions on a bounded time interval with statistically independent coefficients consisting of noncommuting position and momentum operators in a Gaussian quantum state. A similar representation is obtained for the solution of a linear quantum stochastic differential equation which governs the system variables of an open quantum harmonic oscillator. This expansion is applied to computing a quadratic-exponential functional arising as a performance criterion in the framework of risk-sensitive control for this class of open quantum systems.
For a quantum state and a set of observables, we can construct an associated covariance matrix and a natural quantum Fisher information matrix. These two matrices characterize the uncertainty and information content of the observables in the relevant state. An inequality between these two matrices is established. This inequality may be interpreted as a general quantification of the Heisenberg uncertainty principle from a statistical estimation perspective. In particular, it implies a new uncertainty relation which refines the celebrated Schrödinger uncertainty relation.
I Infinite Renormalization of the Hamiltonian Is Necessary.- II Quantum Field Theory Models.- I. The ?22n Model.- Fock space.- Q space.- The Hamiltonian H(g).- Removing the space cutoff.- Lorentz covariance and the Haag-Kastler axioms.- II. The Yukawa Model.- Preliminaries.- First and second order estimates.- Resolvent convergence and self adjointness.- The Heisenberg picture.- III Boson Quantum Field Models.- I. General Results.- Hermite operators.- Gaussian measures and the Schrodinger representation.- Hermite expansions and Fock space.- II. The Solution of Two-Dimensional Boson Models.- The interaction Hamiltonian.- The free Hamiltonian.- Self-adjointness of H(g).- The local algebras and the Lorentz group automorphisms.- IV Boson Quantum Field Models.- III. Further Developments.- Locally normal representations of the observables.- The construction of the physical vaccum.- Formal perturbation theory and models in three space-time dimensions.- V The Particle Structure of the Weakly Coupled P(?)2 Model and Other Applications of High Temperature Expansions.- I. Physics of Quantum Field Models.- Five years of models.- From estimates to physics.- Bound states and resonances.- Phase space localization and renormalization.- VI The Particle Structure of the Weakly Coupled P(?)2 Model and Other Applications of High Temperature Expansions.- II. The Cluster Expansion.- The main results.- The cluster expansion.- Clustering and analyticity: proof of the main results.- Convergence: the main ideas.- An equation of Kirkwood-Salsburg type.- Covariance operators.- Derivatives of covariance operators.- Gaussian integrals.- Convergence: the proof completed.- VII Particles and Bound States and Progess Toward Unitarity and Scaling.- VIII Critical Problems in Quantum Fields.- IX Existence of Phase Transitions for ?24 Quantum Fields.- X Critical Exponents and Renormalization in the ?4 Scaling Limit.- Formulation of the problem.- The scaling and critical point limits.- Renormalization of the ?2(x) field.- Existence of the scaling limit.- The Josephson inequality.- XI A Tutorial Course in Constructive Field Theory.- e?tH as a functional integral.- Examples.- Applications of the functional integral representation.- Ising, Gaussian and scaling limits.- Main results.- Correlation inequalities.- Absence of even bound states.- Bound on g.- Bound on dm2/d? and particles.- The conjecture ?(6) ? 0.- Cluster expansions.- The region of convergence.- The zeroth order expansion.- The primitive expansion.- Factorization and partial resummation.- Typical applications.
For physical systems with Hamiltonians analytic in the position and momentum operators the diagonal matrix elements of the density operator in the coherent state representation may be considered as a phase-space probability measure in the stochastic formulation of quantum mechanics. The correspondence rule relating quantum-mechanical observables to ordinary functions in phase space is obtained. It is shown that the dynamics of the time-dependent phase-space probability densities is again representable in the form of a stochastic process.
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.