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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
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 by identifying the observational noise operator with the initial observable. The initial observable is linked to the quadratic likelihood norm in Hilbert space, which depends on noisy observations and a direct model parameterized by unknown parameters. Using the Karhunen-Loeve expansion, the quantum state is expressed in terms of the eigenvalues and eigenfunctions of the Matern covariance. The likelihood probability is then expressed in terms of the observed state, and the model-dependent state, and is rewritten as a quadratic norm in terms of the quantum states, where the observational noise covariance is proposed to be identified with the initial observable. The main results are summarized by four theorems: the unitarity of the evolution operator, the self-adjointness of the Hamiltonian, the time invariance of the observable, and the equivalence between Heisenberg evolution and Bayesian posterior probability. Limiting cases of the Matern covariance are analyzed as the length parameter tends to zero and infinity. Finally, numerical results support the analytical findings.

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The proposed formulation establishes a formal correspondence between the Heisenberg representation and the Bayesian formulation of inverse problems by identifying the observational noise operator with the initial observable. The initial observable is linked to the quadratic likelihood norm in Hilbert space, which depends on noisy observations and a direct model parameterized by unknown parameters. Using the Karhunen-Loeve expansion, the quantum state is expressed in terms of the eigenvalues and eigenfunctions of the Matern covariance. The likelihood probability is then expressed in terms of the observed state, and the model-dependent state, and is rewritten as a quadratic norm in terms of the quantum states, where the observational noise covariance is proposed to be identified with the initial observable. The main results are summarized by four theorems: the unitarity of the evolution operator, the self-adjointness of the Hamiltonian, the time invariance of the observable, and the equivalence between Heisenberg evolution and Bayesian posterior probability. Limiting cases of the Matern covariance are analyzed as the length parameter tends to zero and infinity. Finally, numerical results support the analytical findings.
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.
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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.
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.
This paper proposes an apparatus-dependent interpretation of self-adjoint realization in boundary-sensitive quantum systems. The main claim is not a new classification theorem for self-adjoint extensions, but a conceptual and operational separation of apparatus dependence into two levels. The first level is structural effective-system selection. In the weighted-endpoint model studied here, this is represented by a model parameter gamma, which specifies the effective Hilbert-space measure or, after unitary transformation, the corresponding inverse-square coupling. The second level is self-adjoint-domain selection after the effective Hilbert space and formal Hamiltonian have already been fixed. In Lebesgue-measure models, this domain choice is represented concretely by a Robin parameter. The paper illustrates this two-level structure through several boundary-sensitive models. A Robin--Dirichlet finite-box model shows that the textbook infinite square well is recovered as the Dirichlet realization, while perfect reflection alone does not uniquely select Dirichlet boundary conditions. A weighted endpoint model distinguishes structural measure selection from residual self-adjoint extension freedom. A thin-wall model gives a concrete asymptotic mechanism by which a resonantly scaled microscopic boundary layer produces an effective Robin condition. Finally, the momentum representation of the Dirichlet box is interpreted as the full-line spectral representation selected by a momentum-measurement setting, rather than as an intrinsic momentum representation of the closed box. The resulting hierarchy is: experimental setting, effective quantum system, state preparation, and measurement statistics. This hierarchy does not modify ordinary quantum measurement theory. It identifies a prior layer that becomes especially relevant in boundary-sensitive systems: before state preparation and measurement are described, the effective quantum system to which the state belongs must be specified.