Entanglement transitions and quantum bifurcations under continuous long-range monitoring

Angelo Russomanno, Giulia Piccitto, and Davide Rossini

Phys. Rev. B 108, 104313 – Published 18 September 2023

Abstract

We study the asymptotic bipartite entanglement entropy of the quantum trajectories of a free-fermionic system, when subject to a continuous nonlocal monitoring. The measurements are described by Gaussian-preserving two-point operators, whose strength decays as a power law with exponent $α$ . Different behaviors of the entanglement entropy with the system size emerge: for $α$ below a given threshold value a volume-law behavior sets in, while for larger $α$ we observe a transition from subvolume to area law, whose exact location depends on the measurements rate and on the presence of a Hamiltonian dynamics. We also consider the expectation probability distribution of the measurement operators, and find that this distribution features a transition from a unimodal to a bimodal shape. We discuss the possible connections between this qualitative change of the distribution and the entanglement transition points.

Received 20 July 2023
Accepted 7 September 2023

DOI:https://doi.org/10.1103/PhysRevB.108.104313

Physics Subject Headings (PhySH)

Entanglement entropy Phase transitions Quantum measurements

Quantum Information, Science & Technology

Authors & Affiliations

Angelo Russomanno ^1,2, Giulia Piccitto ^3,4, and Davide Rossini ³

¹Scuola Superiore Meridionale, Università di Napoli Federico II Largo San Marcellino 10, I-80138 Napoli, Italy
²Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Complesso di Monte S. Angelo, via Cinthia, I-80126 Napoli, Italy
³Dipartimento di Fisica dell'Università di Pisa and INFN, Largo Pontecorvo 3, I-56127 Pisa, Italy
⁴Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria 6, 95125, Catania, Italy

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Issue

Vol. 108, Iss. 10 — 1 September 2023

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Images

Figure 1
Behavior of the average long-time entanglement entropy $S_{N / 2}$ for a system of free fermions, governed by the interplay between the Kitaev-Hamiltonian dynamics and the long-range monitoring. (a) The entanglement entropy $S_{N / 2}$ vs $N$ , for different values of $α$ (increasing $α$ corresponds to a darker color code, as indicated in the legend). (b) $S_{N / 2}$ divided by $N$ vs $α$ , for different sizes (increasing $N$ corresponds to darker markers). (c) $S_{N / 2}$ divided by $ln N$ vs $α$ , for different sizes [same sizes and markers as in panel (b)]. The inset is a magnification of the same data around $α = 3$ . We fix $γ = 0.1$ , $J = 1$ , and $h : 100 \to 0.5$ . Numerical parameters: $Δ t = 5 \times 10^{- 3}$ , $N_{r} \geq 48$ , errorbars as in Appendix pp1.
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Figure 2
Average long-time entanglement entropy for the case of measuring-only dynamics (no Hamiltonian, $J = h = 0$ ). (a) $S_{N / 2} / N$ vs $α$ for different system sizes. (b) $S_{N / 2} / ln N$ versus $α$ for different system sizes. The inset is a magnification of the same data around $α = 2$ . Numerical parameters: $γ Δ t = 5 \cdot 10^{- 4}$ , $N_{r} \geq 48$ , errorbars as in Appendix pp1, same initial state as in Fig. 1.
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Figure 3
Probability distributions of the expectations $ℓ$ of measurement operators, over the sites ( $j$ ) and the discretized time ( $t$ ), for various system sizes (see legend), The four panels correspond to different values of $α = 0$ , 2, 3, 4. Here we fix $γ = 0.1$ , $J = 1$ , and $h : 100 \to 0.5$ . Averages over one single quantum trajectory, evolution up to $T = 10^{4}$ , other numerical parameters as in Fig. 1.
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Figure 4
(a) Absolute value of the position of the maximum of the distributions of the $ℓ$ versus $α$ , for different values of $N$ . The inset shows the variance of the distribution vs $α$ , for different sizes. (b) Absolute maximum of the distribution vs $α$ for different sizes. Same parameters as in Fig. 3.
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Figure 5
Behavior of the entanglement entropy in time for different system sizes (color scale), $γ = 0.1$ , $h = 0.5$ , and $α = 2$ . After a transient, the entanglement entropy saturates to a value that depends on the system size.
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Figure 6
Average long-time entanglement entropy for the case with Hamiltonian ( $J = 1$ , $h = 0.5$ ) and coupling with the environment $γ = 0.5$ . (a) $S_{N / 2} / N$ vs $α$ for different system sizes. The inset shows a comparison of the curves, for fixed size N=64, with $γ = 0.5$ and $γ = 0.1$ . (b) $S_{N / 2} / ln N$ vs $α$ for different system sizes. The inset is a magnification of the same data around $α = 2.5$ . Other numerical parameters and initial state as in Fig. 1.
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