Universality of Entanglement Transitions from Stroboscopic to Continuous Measurements

M. Szyniszewski, A. Romito, and H. Schomerus

Phys. Rev. Lett. 125, 210602 – Published 20 November 2020

Abstract

Measurement-driven transitions between extensive and subextensive scaling of the entanglement entropy receive interest as they illuminate the intricate physics of thermalization and control in open interacting quantum systems. While this transition is well established for stroboscopic measurements in random quantum circuits, a crucial link to physical settings is its extension to continuous observations, where for an integrable model it has been shown that the transition changes its nature and becomes immediate. Here, we demonstrate that the entanglement transition at finite coupling persists if the continuously measured system is randomly nonintegrable, and show that it is smoothly connected to the transition in the stroboscopic models. This provides a bridge between a wide range of experimental settings and the wealth of knowledge accumulated for the latter systems.

Received 12 May 2020
Revised 14 August 2020
Accepted 22 October 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.210602

Physics Subject Headings (PhySH)

Entanglement entropy Entanglement production Quantum chaos Quantum entanglement Quantum measurements Quantum stochastic processes Weak values & weak measurements

1-dimensional spin chains

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

M. Szyniszewski ^1,2,*, A. Romito¹, and H. Schomerus ¹

¹Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
²Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom

^*mszynisz@gmail.com

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 125, Iss. 21 — 20 November 2020

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
We study the entanglement dynamics in a random circuit model, combining unitary evolutions $U$ and measurements $M$ such that one can interpolate between the continuum limit ( $U$ near the identity matrix and measurements weak) and widely studied fully random, stroboscopic models. This is achieved by equipping the unitary matrices with a parameter $μ$ that determines the physical timescale of the dynamics according to $d t \sim μ^{2}$ , and the measurements with a parameter $λ$ so that the effective measurement strength is given by $λ_{0} = λ / μ$ .
Reuse & Permissions
Figure 2
Entanglement dynamics in the random-circuit model of Fig. 1, as captured by the time evolution of the averaged bipartite entanglement entropy $\bar{S}$ . Time is measured in units $t = n μ^{2}$ , the system size is $L = 16$ , and results are averaged over 1000 realizations. The different panels fix the effective measurement strength to (a) $λ_{0} = 0$ , (b) $λ_{0} = 0.3$ , and (c) $λ_{0} = 1.0$ , with the different curves corresponding to different choices of $μ$ . Throughout the whole dynamics, the curves collapse for $μ ≲ 0.1$ , which indicates entering the continuum regime. Increasing the measurement strength suppresses the quasistationary value $S_{\infty}$ , which raises the question of an entanglement transition addressed in the subsequent figures.
Reuse & Permissions
Figure 3
(a) Average saturation entropy $\bar{S_{\infty}}$ and (b) corresponding fluctuations var $S_{\infty}$ , as a function of measurement strength $λ_{0}$ for different system sizes $L$ . The inset in (a) shows $\bar{S_{\infty}}$ for fixed $λ_{0}$ as $L$ is increased, while the inset (b) shows the extrapolation of the position $λ_{0}^{\max}$ of maximal variance to an infinite system size. (c) Tripartite mutual information $I_{3} (A : B : C)$ as a function of measurement strength $λ_{0}$ for different system sizes $L$ , where the subsystems are all of size $L / 4$ , as indicated in the bottom right inset. The left inset focuses on the region where the curves cross, while the top right inset shows the extrapolation of the crossing positions to an infinite system size, where $\bar{L}$ is the average of the system sizes for which the crossing occurs.
Reuse & Permissions
Figure 4
(a) Averaged saturation value $\bar{S_{\infty}}$ of the entanglement entropy for different values of $μ$ in the whole range from the continuum limit ( $μ \to 0)$ to the fully random stroboscopic case ( $μ = 1$ ). In each curve, the measurement strength $λ_{0} = λ / μ$ and the system size are kept fixed. Across the whole range of $μ$ , the entanglement entropy changes its scaling from extensive to subextensive around $λ_{0} \approx 0.3$ . As shown in the subpanels in (b), the qualitative entanglement characteristics in the continuum regime [Figs. 3 and 3] indeed also occur for intermediate values ( $μ = 0.5$ and $μ = 0.7$ ), with $μ = 1$ reproducing the conventional stroboscopic case.
Reuse & Permissions

Physical Review Letters