Information dynamics and symmetry breaking in generic monitored $\mathbb{Z}_2$-symmetric open quantum systems

TL;DR

This paper explores the phases of monitored open quantum systems with $ ext{Z}_2$ symmetry, revealing how information is retained, leaked, or learned, and connects these phases to classical Ising models.

Contribution

It systematically characterizes symmetry-breaking phases in monitored quantum systems and links them to classical statistical models, including a self-dual Ising model at criticality.

Findings

Identifies three distinct information phases: fully retained, leaked, and learned.

Establishes a connection between quantum phases and classical Ising models with negative weights.

Provides numerical evidence using tensor network simulations for the phase structure.

Abstract

We investigate the steady-state phases of generic $\mathbb{Z}_2$-symmetric monitored, open quantum dynamics. We describe the phases systematically in terms of both information-theoretic diagnostics and spontaneous breaking of strong and weak symmetries of the dynamics. We find a completely broken phase where information is retained by the quantum system, a strong-to-weak broken phase where information is leaked to the environment, and an unbroken phase where information is learned by the observer. We find that weak measurement and dephasing alone constitute a minimal model for generic open systems with $\mathbb{Z}_2$ symmetry, but we also explore perturbations by unitary gates. For a 1d set of qubits, we examine information-theoretic and symmetry-breaking observables in the path integral of the doubled state. This path integral reduces to the standard classical 2d random-bond Ising model in certain limits but generically involves negative weights, enabling a special self-dual random-bond Ising model at the critical point when only measurements are present. We obtain numerical evidence for the steady-state phases using efficient tensor network simulations of the doubled state.