The Bayesian inverse of a quantum channel $\mathcal{E}$ is a channel $\mathcal{F}$ in the reverse direction of $\mathcal{E}$ that yields time-symmetric correlations for sequential measurements performed on open quantum systems. Such an operational form of time-reversal symmetry for open quantum systems is quite remarkable, as the dynamics of open quantum systems are inherently irreversible due to system-environment interactions. Similar to the Petz map, a Bayesian inverse $\mathcal{F}$ is defined with respect to a fiducial reference state $ρ$ for the channel $\mathcal{E}$ . However, Bayesian inverses do not always exist, and it is often a non-trivial task to determine the set of states $ρ$ for which a Bayesian inverse of $\mathcal{E}$ exists. In this work, we solve the general problem of quantum Bayesian inversion for unital channels acting on a single qubit. Our analysis is streamlined by demonstrating that finding a Bayesian inverse for a unital qubit channel may be reduced to finding a Bayesian inverse of a Pauli channel, which is simply a mixture of unitary channels associated with the Pauli matrices. As such, we provide a complete description of when operational time-reversal symmetry is attainable for sequential measurements of a single qubit in the presence of unital noise.The Bayesian inverse of a quantum channel $\mathcal{E}$ is a channel $\mathcal{F}$ in the reverse direction of $\mathcal{E}$ that yields time-symmetric correlations for sequential measurements performed on open quantum systems. Such an operational form of time-reversal symmetry for open quantum systems is quite remarkable, as the dynamics of open quantum systems are inherently irreversible due to system-environment interactions. Similar to the Petz map, a Bayesian inverse $\mathcal{F}$ is defined with respect to a fiducial reference state $ρ$ for the channel $\mathcal{E}$. However, Bayesian inverses do not always exist, and it is often a non-trivial task to determine the set of states $ρ$ for which a Bayesian inverse of $\mathcal{E}$ exists. In this work, we solve the general problem of quantum Bayesian inversion for unital channels acting on a single qubit. Our analysis is streamlined by demonstrating that finding a Bayesian inverse for a unital qubit channel may be reduced to finding a Bayesian inverse of a Pauli channel, which is simply a mixture of unitary channels associated with the Pauli matrices. As such, we provide a complete description of when operational time-reversal symmetry is attainable for sequential measurements of a single qubit in the presence of unital noise.

quant-pharXiv:2605.10375May 12, 20260 citations

Bayesian Inverse Quantum Theory

Jörg C. Lemm, Joerg Uhlig

. A Bayesian approach is developed to determine quantum mechanical potentials from empirical data. Bayesian methods, combining empirical measurements and a priori information, provide exible tools for such empirical learning problems. The paper presents the basic theory, concentrating in particular on measurements of particle coordinates in quantum mechanical systems at nite temperature. The computational feasibility of the approach is demonstrated by numerical case studies. Finally, it is shown how the approach can be generalized to such many{body and few{body systems for which a mean eld description is appropriate. This is done by means of a Bayesian inverse Hartree{Fock approximation. Contents 1 Introduction 2 2 The Bayesian approach 3 2.1 Basic notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 An Example: Regression . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Inverse quantum statistics 7 3.1 The likelihood model of quantum theory . . . . . . . ..... A Bayesian approach is developed to determine quantum mechanical potentials from empirical data. Bayesian methods, combining empirical measurements and a priori information, provide exible tools for such empirical learning problems. The paper presents the basic theory, concentrating in particular on measurements of particle coordinates in quantum mechanical systems at nite temperature. The computational feasibility of the approach is demonstrated by numerical case studies. Finally, it is shown how the approach can be generalized to such many{body and few{body systems for which a mean eld description is appropriate. This is done by means of a Bayesian inverse Hartree{Fock approximation. Contents 1 Introduction 2 2 The Bayesian approach 3 2.1 Basic notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 An Example: Regression . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Inverse quantum statistics 7 3.1 The likelihood model of quantum theory . . . . . . . ....

Jan 1, 20004 citations

The Bayesian inverse of a quantum channel

\mathcal{E}

is a channel

\mathcal{F}

in the reverse direction of

\mathcal{E}

that yields time-symmetric correlations for sequential measurements performed on open quantum systems. Such an operational form of time-reversal symmetry for open quantum systems is quite remarkable, as the dynamics of open quantum systems are inherently irreversible due to system-environment interactions. Similar to the Petz map, a Bayesian inverse

\mathcal{F}

is defined with respect to a fiducial reference state

ρ

for the channel

\mathcal{E}

. However, Bayesian inverses do not always exist, and it is often a non-trivial task to determine the set of states

ρ

for which a Bayesian inverse of

\mathcal{E}

exists. In this work, we solve the general problem of quantum Bayesian inversion for unital channels acting on a single qubit. Our analysis is streamlined by demonstrating that finding a Bayesian inverse for a unital qubit channel may be reduced to finding a Bayesian inverse of a Pauli channel, which is simply a mixture of unitary channels associated with the Pauli matrices. As such, we provide a complete description of when operational time-reversal symmetry is attainable for sequential measurements of a single qubit in the presence of unital noise.