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

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.

Figures (14)

• Figure 1: Generic open quantum systems evolve internally (unitary), couple to an observer (measurement), and couple to a bath (decoherence). We are interested in the steady-state phases of such systems. Each pair of these ingredients is well-studied: unitaries and measurements drive MIPTs, unitaries and decoherence are described by Lindbladians, and measurements and decoherence model QEC. We find that the steady-state phases of generic open quantum systems can be organized in terms of symmetry breaking and information theory.
• Figure 2: (a) The circuit architecture we consider and (b) individual circuit components. The dynamics comprise alternating layers of Pauli-X weak measurements (red) and dephasing (yellow), and Pauli-ZZ weak measurements (blue) and dephasing (yellow). We emphasize that the dynamics may be viewed in terms of the doubled state $| \rho \rangle\rangle$, which has forward and backward degrees of freedom. Measurements act separately on the degrees of freedom (illustrated by two decoupled gates) and dephasing acts on both degrees of freedom simultaneously (illustrated by a cubic gate each).
• Figure 3: Phase diagram for the staggered XXZ chain (quantum Ashkin-Teller model). The forced measurement dynamics and the $2$-replica dynamics explore this phase diagram according to Eqs. \ref{['eq:forced_parameters_1']} to \ref{['eq:forced_parameters_3']} and Eqs. \ref{['eq:replica_parameters_1']} to \ref{['eq:replica_parameters_3']}, respectively. Consequently, both models see the phase diagram above the dashed line at $K/J = 1$, but only the forced measurement dynamics explores the shaded region below this line. Critical points of particular interest are the $SU(2)$-invariant point at $\Delta/J = 0$ and the critical points at $\Delta/J = \pm 1$ where the model can be reduced from $2L$ spins down to $L$ spins.
• Figure 4: Schematic phase diagram for the intrinsic phases of our dynamics. Phase 1 is identified by $\overline{I_c} = 1$, indicating a memory, and $\overline{\kappa_{EA}} \propto L$, indicating SSB of the weak symmetry. Phase 3 is identified by $\overline{S_R} = 0$, indicating learning, and $\overline{\kappa_2} \propto 1$, indicating that the strong symmetry is unbroken. Phase 2 is the complement of these two phases, where neither a memory nor learning are possible, and where strong-to-weak SSB is present. $\lambda$ controls the relative strength of Pauli-X and Pauli-ZZ measurements, with $\lambda_x = \delta \lambda$ and $\lambda_{zz} = \delta(1-\lambda)$ where we have selected $\delta = 0.7$. $q_x = q_{zz} = q$ controls the strength of decoherence.
• Figure 5: Observables for the phase transitions at $q = 0.1$. (a) The Edwards-Anderson susceptibility diverges in Phase 1 where the $\mathbb{Z}_2$ symmetry is completely broken. (b) The Rényi-$2$ coherent information undergoes a transition between Phase 1 and Phase 2 at $\lambda_{c,{12}} \approx 0.265$. The data exhibit a scaling collapse with $\nu_{12} \approx 1.2$; however, we do not expect this to reflect the underlying criticality that would be diagnosed by the standard (Rényi-$1$) coherent information. (c) The Rényi-$2$ susceptibility diverges in Phases 1 and 2 where the strong $\mathbb{Z}_2$ symmetry is broken down to a weak symmetry. (d) The reference entropy undergoes a transition between Phase 2 and Phase 3 at $\lambda_{c,{23}} \approx 0.725$. The data exhibit a scaling collapse with $\nu_{23} = 1.5$, consistent with RBIM criticality along the Nishimori line PhysRevB.111.094201.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof