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arXiv:2206.12429 · Physical Review Letters

Transitions in the Learnability of Global Charges from Local Measurements

Fergus Barratt, Utkarsh Agrawal, Andrew C. Potter, Sarang Gopalakrishnan, Romain Vasseur

Published Nov 10, 2022• 61 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.129.200602

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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.
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 study the critical properties of random quantum circuits with a U(1)U(1)U(1) symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored U(1)U(1)U(1)-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored U(1)U(1)U(1)-symmetric Haar and stabilizer random circuits.
Charge-learnability transitions in monitored symmetric quantum circuits reveal how local measurement records acquire sufficient information to infer a conserved charge. Here we extend charge learnability to probabilistic weak measurements, for which the measurement probability and measurement strength are independently tunable. We find that the learnability phase boundary is organized by the informational power of local measurement. We further introduce cross entropy as a label-sensitive diagnostic that distinguishes unbiased, biased, and antibiased decoder variants. Finally, the exact record--label mutual information provides a decoder-independent benchmark for the information fundamentally available for charge inference. Our results establish informational power of local measurement as a unifying principle for charge learnability under general monitoring protocols.
We study the critical properties of random quantum circuits with a U(1)U(1)U(1) symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored U(1)U(1)U(1)-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored U(1)U(1)U(1)-symmetric Haar and stabilizer random circuits.
We consider how the presence of conserved charges affects memory in a classical stochastic process, the symmetric exclusion process, with an observer constantly measuring a single site. We find that the observer's measurement record becomes Markovian (i.e., loses memory) on a timescale that depends on their knowledge of the global charge, namely the total particle number. In particular, when the global charge is unknown a priori, the observer's time series Markovianizes on a timescale constrained by their ability to learn it from their measurement record. Augmenting the observer's record with bulk measurements drives a charge-learnability transition between charge-fuzzy and -sharp phases. We show that the memory timescale tracks the learnability timescale, diverging in the fuzzy phase and remaining finite in the sharp phase.
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