Controlling Entanglement at Absorbing State Phase Transitions in Random Circuits

Piotr Sierant and Xhek Turkeshi

Phys. Rev. Lett. 130, 120402 – Published 23 March 2023

Abstract

Many-body unitary dynamics interspersed with repeated measurements display a rich phenomenology hallmarked by measurement-induced phase transitions. Employing feedback-control operations that steer the dynamics toward an absorbing state, we study the entanglement entropy behavior at the absorbing state phase transition. For short-range control operations, we observe a transition between phases with distinct subextensive scalings of entanglement entropy. In contrast, the system undergoes a transition between volume-law and area-law phases for long-range feedback operations. The fluctuations of entanglement entropy and of the order parameter of the absorbing state transition are fully coupled for sufficiently strongly entangling feedback operations. In that case, entanglement entropy inherits the universal dynamics of the absorbing state transition. This is, however, not the case for arbitrary control operations, and the two transitions are generally distinct. We quantitatively support our results by introducing a framework based on stabilizer circuits with classical flag labels. Our results shed new light on the problem of observability of measurement-induced phase transitions.

Received 6 January 2023
Revised 23 February 2023
Accepted 28 February 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.120402

Physics Subject Headings (PhySH)

Dynamical phase transitions Quantum circuits Quantum control Quantum entanglement Quantum feedback Quantum phase transitions

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Piotr Sierant ¹ and Xhek Turkeshi ²

¹ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
²JEIP, USR 3573 CNRS, Collège de France, PSL Research University, 11 Place Marcelin Berthelot, 75321 Paris Cedex 05, France

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Vol. 130, Iss. 12 — 24 March 2023

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Images

Figure 1
(a) A single layer of our quantum circuit consists of measurement operations (in gray), short-range unitary gates $U_{i, i + 1}$ (in blue), and a variable-range control operation $A (α)$ . Both $U_{i, i + 1}$ and $A (α)$ are conditioned on measurement outcomes which lead to the appearance of ordered domains [highlighted in orange in (b) and (c)] and disordered regions in the state of the system; $A (α)$ generates entanglement (represented by blue lines) locally, (b), or entangles distant disordered regions, (c), depending on the range $α$ .
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Figure 2
(a) The entanglement entropy $S_{X}$ and the order parameter $O$ (see text) at time $t = 4 L$ indicate, for $α = 3.5$ , the presence of a volume-law, an area-law nonabsorbing, and an area-law absorbing phase. (b) For long-range interactions, $α = 0.5$ , we observe a direct transition between volume-law and area-law phases that coincides with the APT at $p = p_{c}^{APT}$ . (c) Dynamical phases of the circuit as a function of $α$ and the measurement rate $p$ . The critical points $p_{c}^{MIPT}$ for the flagged Clifford circuits are denoted in blue.
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Figure 3
Average defect density $n_{def}$ (a) and entanglement entropy $S$ in the short-range (long-range) regime (b) [(c)] close to the APT. Darker colors reflect larger system sizes in the range $L = 128 \div 1024$ , while the red dashed lines indicate the DP universality. In (a), the bottom inset illustrates the development of the DP critical exponent $δ_{DP} ≃ 0.1595 (1)$ , while the top right inset shows a data collapse with $δ_{n} = 0.159 (1)$ , $p_{c} = p_{c}^{APT} = 0.6726 (2)$ , and $ν_{∥} = 1.73 (1)$ . Similarly, (b) and (c) present the behavior of $δ_{S} (t)$ for an area-to-area and a volume-to-area transition for subsystem size $l_{X} = 8$ , respectively. In (b), the collapse is for $δ_{S} = 0.16 (1)$ , $p_{c} = 0.6722 (5)$ , and $ν_{∥} = 1.73 (8)$ . In (c), the critical values are $δ_{S} = 0.16 (1)$ , $p_{c} = 0.672 (1)$ , and $ν_{∥} = 1.70 (5)$ . The top left insets in (b) and (c) show the saturation value $S_{sat}$ of $S (t)$ for $p < p_{c}^{APT}$ with $l_{X}$ . For $α = 3.5$ , $S_{sat}$ becomes independent of $l_{X}$ , whereas for $α = 0.5$ , $S_{sat} \sim l_{X}$ . Our results are averaged over no less than $10^{3}$ circuit realizations.
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Figure 4
(a) Times $τ_{⋆}$ of the start of decay to the absorbing state at system size $L$ . For $p < p_{c}^{APT}$ , $τ_{⋆}$ is exponentially diverging with $L$ , whereas for $p > p_{c}^{APT}$ it saturates with $L$ . (b) Exponent $δ_{S}$ for a subsystem size $l_{X} = 64$ ; cf. Fig. 3. In (c),(d), we present $δ_{S}$ for $l_{X} = 8$ , 64 for a different protocol $A_{2}^{α}$ ; see text. $δ_{S}$ saturates to value increasing with $l_{X}$ , denoted by the dashed brown lines, $δ = 0.255$ in (c) and $δ = 0.3$ in (d). This behavior is distinct from the $A_{1}^{α}$ case. In (b)–(d), darker colors refer to larger system sizes in the range $L = 128 \div 1024$ , and the dashed red line is the DP reference $δ = 0.1595$ .
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