Transitions in the learnability of global charges from local measurements

TL;DR

This paper investigates how the ability to learn a global conserved charge in monitored quantum systems from local measurements changes with measurement rate, revealing phase transitions in learnability depending on available information.

Contribution

It introduces an optimal classical classifier for reconstructing global charges from local measurements in symmetric quantum circuits and demonstrates phase transitions in classifier performance.

Findings

Phase transitions in learnability as a function of measurement rate.

Existence of an optimal classifier for charge reconstruction.

Numerical evidence of performance changes with additional knowledge.

Abstract

We consider monitored quantum systems with a global conserved charge, and ask how efficiently an observer ("eavesdropper") can learn the global charge of such systems from local projective measurements. We find phase transitions as a function of the measurement rate, depending on how much information about the quantum dynamics the eavesdropper has access to. For random unitary circuits with U(1) symmetry, we present an optimal classical classifier to reconstruct the global charge from local measurement outcomes only. We demonstrate the existence of phase transitions in the performance of this classifier in the thermodynamic limit. We also study numerically improved classifiers by including some knowledge about the unitary gates pattern.