A process that images or measures bond energies in the critical Ising model can be in distinct measurement ``phases'', depending on the precision of measurement. We study the transition into the strong-measurement phase using replica field theory (an epsilon expansion around six dimensions) and numerical simulations in two dimensions. The results reveal multiscaling of correlation functions at the critical point, and a striking enlarged symmetry of the replica description. This is an analog of the Nishimori phenomenon in the Ising spin glass, in a distinct replica limit. The enlarged symmetry is present microscopically for certain measurement protocols, but more generally can emerge in the infrared, and it fixes the exact value of the exponent for the Edwards-Anderson correlator both in 2D and near the upper critical dimension. We also examine the epsilon expansion for models with power-law interactions and/or long-range measurement.A process that images or measures bond energies in the critical Ising model can be in distinct measurement ``phases'', depending on the precision of measurement. We study the transition into the strong-measurement phase using replica field theory (an epsilon expansion around six dimensions) and numerical simulations in two dimensions. The results reveal multiscaling of correlation functions at the critical point, and a striking enlarged symmetry of the replica description. This is an analog of the Nishimori phenomenon in the Ising spin glass, in a distinct replica limit. The enlarged symmetry is present microscopically for certain measurement protocols, but more generally can emerge in the infrared, and it fixes the exact value of the exponent for the Edwards-Anderson correlator both in 2D and near the upper critical dimension. We also examine the epsilon expansion for models with power-law interactions and/or long-range measurement.

cond-mat.stat-mechCross-listedarXiv:2604.23346Apr 28, 20260 citations

Operational time-reversal symmetry for unital qubit channels

Ouyang Ting, James Fullwood, Zhen Wu

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

Time Symmetry and the Theory of Measurement

K. Datta

Springer eBooksJan 1, 19860 citations

Time Symmetry in the Quantum Process of Measurement

Yakir Aharonov, Peter G. Bergmann, Joel L. Lebowitz

WORLD SCIENTIFIC eBooksAug 1, 20250 citations

Time symmetry and quantum measurements

Michel Bitbol

Physics Letters AMay 1, 19863 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.