Operational time-reversal symmetry for unital qubit channels

• URL: http://arxiv.org/abs/2605.10375v1
• Date: Mon, 11 May 2026 11:19:41 GMT
• Title: Operational time-reversal symmetry for unital qubit channels
• Abstract summary: A Bayesian inverse of a quantum channel $mathcalE$ is a channel $mathcalF$ in the reverse direction of $mathcalE$.<n>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.
• Score: 4.650614816375192
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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