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arXiv:2503.10308 · Physical review. B./Physical review. B

Mixed-state learnability transitions in monitored noisy quantum dynamics

Hansveer Singh, Romain Vasseur, Andrew C. Potter, Sarang Gopalakrishnan

Published Jan 12, 2026• 2 citations
arXivHTMLPDF

Abstract

We consider learnability transitions in monitored quantum systems that undergo noisy evolution, subject to a global strong symmetry—i.e., in addition to the measuring apparatus, the system can interact with an unobserved environment, but does not exchange charge with it. As in the pure-state setting, we find two information-theoretic phases—a sharp (fuzzy) phase in which an eavesdropper can rapidly (slowly) learn the symmetry charge. However, because the dynamics is noisy, both phases can be simulated efficiently using tensor networks. Indeed, even when the true dynamics is unitary, introducing noise by hand allows an eavesdropper to efficiently learn the symmetry charge from local measurements, as we demonstrate. We identify the fuzzy phase in this setting as a mixed-state phase that exhibits spontaneous strong-to-weak symmetry breaking.

DOI
10.1103/6b9y-scyz

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We consider learnability transitions in monitored quantum systems that undergo noisy evolution, subject to a global strong symmetry—i.e., in addition to the measuring apparatus, the system can interact with an unobserved environment, but does not exchange charge with it. As in the pure-state setting, we find two information-theoretic phases—a sharp (fuzzy) phase in which an eavesdropper can rapidly (slowly) learn the symmetry charge. However, because the dynamics is noisy, both phases can be simulated efficiently using tensor networks. Indeed, even when the true dynamics is unitary, introducing noise by hand allows an eavesdropper to efficiently learn the symmetry charge from local measurements, as we demonstrate. We identify the fuzzy phase in this setting as a mixed-state phase that exhibits spontaneous strong-to-weak symmetry breaking.
We explore the information-theoretic phases of monitored quantum circuits subject to dynamics that conserves both charge and dipole moment, as well as measurements of the local charge density. Explicitly, both charge and dipole-moment conservation are strong symmetries, but under the dynamics they can be spontaneously broken to weak symmetries: this spontaneous symmetry breaking has an information-theoretic interpretation in terms of whether one can learn global charges from local measurements. We find a rich phase diagram: in one spatial dimension, charge is always easy to learn, while dipole moment can be either easy or hard. In two dimensions, we find three phases: for frequent measurements, both charge and dipole moment are easy to learn; as the measurement rate is decreased, first dipole moment and then charge become hard. In two dimensions, the low-measurement phase is an exotic critical phase with anisotropic spacetime scaling, analogous to a smectic liquid crystal.
Information dynamics and symmetry breaking in generic monitored Z2\mathbb{Z}_2Z2​-symmetric open quantum systems
We investigate the steady-state phases of generic Z2\mathbb{Z}_2Z2​-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 Z2\mathbb{Z}_2Z2​ 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.
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. We also study numerically improved classifiers by including some knowledge about the unitary gates pattern.
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Monitored quantum dynamics—unitary evolution interspersed with measurements—has recently emerged as a rich domain for phase structure in quantum many-body systems away from equilibrium. Here we study monitored dynamics from the point of view of an eavesdropper who has access to the classical measurement outcomes, but not to the quantum many-body system. We show that a measure of information flow from the quantum system to the classical measurement record—the —undergoes a phase transition in correspondence with the measurement-induced phase transition (MIPT). This transition determines the eavesdropper’s (in)ability to learn properties of an unknown initial quantum state of the system, given a complete classical description of the monitored dynamics and arbitrary classical computational resources. We make this learnability transition concrete by defining classical shadow protocols that the eavesdropper may apply to this problem, and show that the MIPT manifests as a transition in the sample complexity of various shadow-estimation tasks, which become harder in the low-measurement phase. We focus on three applications of interest: Pauli expectation values (where we find the MIPT appears as a point of optimal learnability for typical Pauli operators), many-body fidelity, and global charge in U ( 1 ) -symmetric dynamics. Our work unifies different manifestations of the MIPT under the umbrella of and gives this notion a general operational meaning via classical shadows. Published by the American Physical Society 2024
Monitored many-body quantum systems can undergo sharp learnability transitions characterized by how much information can be learned by the observer. When the dynamics conserves a non-Abelian charge, such as an SU(2)SU(2)SU(2) spin, understanding how the observer learns the total charge remains an outstanding problem. Unlike the Abelian case, where charge measurements on distinct sites commute, the SU(2)SU(2)SU(2)-symmetric readouts are noncommuting fusion measurements, making learning a genuinely quantum inference problem. In this work, we propose a theory of 1+1d1+1d1+1d monitored quantum dynamics with SU(2)SU(2)SU(2) symmetry, and show that it can be described by an effective replicated loop model comprised of a replica-pairing field and a diffusive (z=2z=2z=2) background sector that carries the SU(2)SU(2)SU(2) charge and remains gapless throughout the phase diagram. Our theory predicts that the "spin-sharpening'' and entanglement transitions coincide as a single transition. Ordering of the pairing field produces volume-law entanglement and hides the background sector from measurements, leading to a learning time of t∼L3t\sim L^{3}t∼L3 for the total spin. When the pairing field disorders, the background sector alone gives logarithmic entanglement and a diffusive learning time t∼L2t\sim L^{2}t∼L2. Our analysis is controlled by a large-loop-fugacity expansion.