Transitions in the learnability of global charges from local measurements

Understand

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.