Description
Ref. [1] introduced measurement-induced charge-sharpening in quantum circuits with qubits undergoing $U(1)$ dynamics alongside 'bath' qudits. Here the charge variance of the density matrix undergoes a (purportedly KT) transition before the entanglement transition. In the $d \rightarrow \infty$ limit, the transition can be described by stat-mech model of constrained hard-core walkers.
We show in this limit, the model can be framed as a Bayesian inference on a hidden Markov model, where a 'student' tries to infer state of the 'teacher' given noisy measurements. In this setting a concept of Bayes non-optimality can be invoked, referring to when the parameters (such as signal-to-noise ratio) assumed by the student differs from that of the teacher. The Bayes optimal case corresponds to equal student and teacher's parameters, and to the Born rule. In the spin-glass literature, it is also referred to as the Nishimori line. When the teacher supplies random noise but the student assumes some signal, this corresponds to the 'forced measurement' limit [2]. Measurement-free circuit in the quantum setting corresponds to the student assuming that there is zero information in the measurements.
References
[1]: Agrawal, U., Zabalo, A., Chen, K., Wilson, J. H., Potter, A. C., Pixley, J. H., ... & Vasseur, R. (2022). Entanglement and charge-sharpening transitions in U (1) symmetric monitored quantum circuits. Physical Review X, 12(4), 041002.
[2]: Feng, X., Skinner, B., & Nahum, A. (2023). Measurement-induced phase transitions on dynamical quantum trees. PRX Quantum, 4(3), 030333.