Asymptotic Fate of Continuously Monitored Quantum Systems

TL;DR

This paper characterizes the long-term behavior of finite-dimensional quantum systems under continuous measurement, revealing spontaneous localization transitions, symmetry violations, and probabilistic state collapses with implications for quantum state stabilization.

Contribution

It provides a complete characterization of asymptotic dynamics, introduces a generalized update rule with a Born-like probability assignment, and offers algorithms to identify invariant subspaces and stationary states.

Findings

Localization transitions occur on individual trajectories.

Symmetries are broken during localization, violating ergodicity.

Algorithms for identifying invariant subspaces and stationary states.

Abstract

A quantum trajectory is the natural response of a quantum system subject to external perturbations due to continuous indirect measurement. We completely characterize the asymptotic behavior of continuously monitored quantum systems in finite dimensions and show that generically, spontaneous irreversible localization transitions on the level of individual realizations occur, where the evolution becomes effectively constrained to one of the irreducible components of the total Hilbert space. More generally, localization can be either complete, where the strongest possible confinement is achieved, or incomplete, where localization terminates prematurely. On the trajectory level, symmetries and conserved quantities are no longer respected and localization transitions occur concurrently with violations of ergodicity. As a result, a generalized update rule emerges that effectively projects the system onto one of several possible time evolutions. The update comes equipped with a generalized Born rule that assigns probabilities to these irreversible events. Spontaneous transitions thus occur probabilistically and can deviate considerably from the behavior of the ensemble. In particular, time and ensemble average no longer commute which gives rise to global violations of ergodicity, while on a local level, ergodicity is restored. We explicitly illustrate the asymptotic behavior of continuously monitored systems in a series of examples demonstrating stabilization of many-body scar states and generation of Bell states from local measurement. Finally, we present two algorithms, one based on simultaneous block diagonalization and one based on quantum trajectories to identify all the minimal orthogonal subspaces of the Lindblad equation and all the extremal stationary states which are supported on them.