Random-matrix models of monitored quantum circuits

TL;DR

This paper analyzes the statistical properties of Kraus operators in monitored quantum circuits, revealing how measurements influence quantum dynamics and entanglement, with insights applicable to broader quantum systems.

Contribution

It provides analytical results for the distribution of Kraus operators in monitored circuits, including a solvable Fokker-Planck equation for weak measurements, extending random matrix theory to quantum monitoring.

Findings

Distribution of Born probabilities is log-normal at long times.

Derived an exact Fokker-Planck equation for singular values of Kraus operators.

Generalized Porter-Thomas distribution to monitored quantum circuits.

Abstract

We study the competition between Haar-random unitary dynamics and measurements for unstructured systems of qubits. For projective measurements, we derive various properties of the statistical ensemble of Kraus operators analytically, including the purification time and the distribution of Born probabilities. The latter generalizes the Porter-Thomas distribution for random unitary circuits to the monitored setting and is log-normal at long times. We also consider weak measurements that interpolate between identity quantum channels and projective measurements. In this setting, we derive an exactly solvable Fokker-Planck equation for the joint distribution of singular values of Kraus operators, analogous to the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation modelling disordered quantum wires. We expect that the statistical properties of Kraus operators we have established for these simple systems will serve as a model for the entangling phase of monitored quantum systems more generally.