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Physical review. B./Physical review. B

Universality of the cross-entropy in Z 2 symmetric monitored quantum circuits

Maria Tikhanovskaya, Ali Lavasani, Matthew P. A. Fisher, Sagar Vijay

Published Jun 25, 2024• 9 citations
Publisher

Abstract

In monitored quantum many-body dynamics, information about the evolving state can ``leak'' into the measurement record, as quantified by the linear-cross entropy (LXE). Here, the authors show that the LXE serves as an order parameter that also distinguishes various area-law-entangled phases of monitored circuits with a global Ising symmetry. The behavior of the LXE at a critical point separating these phases quantitatively agrees with predictions from the conformal field theory describing this critical point. This connection between out-of-equilibrium dynamics and equilibrium statistical mechanics suggests the possibility of the LXE as a probe of universal critical behavior in other kinds of monitored quantum evolution.

DOI
10.1103/physrevb.109.224313

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Italian Quantum Information Science Conference (IQIS)•Jun 3, 2019•0 citations
In monitored quantum many-body dynamics, information about the evolving state can ``leak'' into the measurement record, as quantified by the linear-cross entropy (LXE). Here, the authors show that the LXE serves as an order parameter that also distinguishes various area-law-entangled phases of monitored circuits with a global Ising symmetry. The behavior of the LXE at a critical point separating these phases quantitatively agrees with predictions from the conformal field theory describing this critical point. This connection between out-of-equilibrium dynamics and equilibrium statistical mechanics suggests the possibility of the LXE as a probe of universal critical behavior in other kinds of monitored quantum evolution.
Universality of the cross entropy in Z2\mathbb{Z}_2Z2​ symmetric monitored quantum circuits
The linear cross-entropy (LXE) has been recently proposed as a scalable probe of the measurement-driven phase transition between volume- and area-law-entangled phases of pure-state trajectories in certain monitored quantum circuits. Here, we demonstrate that the LXE can distinguish distinct area-law-entangled phases of monitored circuits with symmetries, and extract universal behavior at the critical points separating these phases. We focus on (1+1)-dimensional monitored circuits with an on-site Z2\mathbb{Z}_{2}Z2​ symmetry. For an appropriate choice of initial states, the LXE distinguishes the area-law-entangled spin glass and paramagnetic phases of the monitored trajectories. At the critical point, described by two-dimensional percolation, the LXE exhibits universal behavior which depends sensitively on boundary conditions, and the choice of initial states. With open boundary conditions, we show that the LXE relates to crossing probabilities in critical percolation, and is thus given by a known universal function of the aspect ratio of the dynamics, which quantitatively agrees with numerical studies of the LXE at criticality. The LXE probes correlations of other operators in percolation with periodic boundary conditions. We show that the LXE is sensitive to the richer phase diagram of the circuit model in the presence of symmmetric unitary gates. Lastly, we consider the effect of noise during the circuit evolution, and propose potential solutions to counter it.
Quantum circuits offer a versatile platform for simulating digital quantum dynamics and uncovering novel states of nonequilibrium quantum matter. One principal example are measurement-induced phase transitions arising from nonunitary dynamics in monitored circuits, which employ midcircuit measurements as an essential building block next to standard unitary gates. Although a comprehensive understanding of the dynamics in generic circuits is still evolving, we contend that monitored quantum circuits give rise to robust phases of dynamic matter, which---akin to Hamiltonian ground-state phases---yield emergent universal behavior, which can be categorized based on circuit symmetries and spatial dimensionality. To illustrate this concept, we focus on measurement-only quantum circuits within symmetry classes BDI and D, which are measurement-only circuit adaptations of the paradigmatic Kitaev and Yao-Kivelson models, embodying particle-hole-symmetric Majorana fermions with or without time reversal. We establish a general framework (Majorana loop models) for both symmetry classes (in arbitrary spatial dimensions) to provide access to the phenomenology of the entanglement dynamics in these circuits, displaying both an area-law phase of localized Majorana loops and a delocalized, highly entangled Majorana liquid phase. The two phases are separated by a continuous transition displaying quantum Lifshitz scaling, albeit with critical exponents of two distinct, non-Hamiltonian universality classes. The loop model framework provides not only analytical understanding of these universality classes in terms of nonlinear sigma models but also allows for highly efficient numerical techniques capable of simulating excessively large circuits with up to 108{10}^{8}108 qubits. We utilize this framework to accurately determine universal probes that distinguish both the entangled phases and the critical points of the two symmetry classes. Our work thereby further solidifies the concept of emergent circuit phases and their phase transitions.
We propose that the quantum conditional mutual information (QCMI), computed with a suitably chosen partition of the system, serves as a powerful probe for detecting measurement-induced entanglement phase transitions in monitored quantum circuits. To demonstrate this, we investigate monitored variable-range Clifford circuits and identify the phase boundary between volume-law and area-law entanglement phases by performing finite-size scaling analyses of the QCMI. Assuming that the entanglement entropy exhibits a logarithmic dependence on system size at criticality in short-range interacting cases, we further show that the QCMI allows for the simultaneous determination of both the critical point and the universal coefficient of the logarithmic term in the entanglement entropy via a crossing-point analysis. For the shortest-range interacting case studied, we obtain the thermodynamic-limit value of the coefficient as c~=1.519(3)\tilde{c}=1.519(3)c~=1.519(3), which is significantly smaller than values reported in previous studies.
We investigate entanglement phase transitions from volume-law to area-law entanglement in a quantum many-body state under continuous position measurement on the basis of the quantum trajectory approach. We find the signatures of the transitions as peak structures in the mutual information as a function of measurement strength, as previously reported for random unitary circuits with projective measurements. At the transition points, the entanglement entropy scales logarithmically and various physical quantities scale algebraically, implying emergent conformal criticality, for both integrable and nonintegrable one-dimensional interacting Hamiltonians; however, such transitions have been argued to be absent in noninteracting regimes in some previous studies. With the aid of U(1)U(1)U(1) symmetry in our model, the measurement-induced criticality exhibits a spectral signature resembling a Tomonaga-Luttinger liquid theory from symmetry-resolved entanglement. These intriguing critical phenomena are unique to steady-state regimes of the conditional dynamics at the single-trajectory level and are absent in the unconditional dynamics obeying the Lindblad master equation, in which the system ends up with the featureless, infinite-temperature mixed state. We also propose a possible experimental setup to test the predicted entanglement transition based on the subsystem particle-number fluctuations. This quantity should readily be measured by the current techniques of quantum gas microscopy and is in practice easier to obtain than the entanglement entropy itself.
Monitored quantum circuits can exhibit an entanglement transition as a function of the rate of measurements, stemming from the competition between scrambling unitary dynamics and disentangling projective measurements.We study how entanglement dynamics in nonunitary quantum circuits can be enriched in the presence of charge conservation, using a combination of exact numerics and a mapping onto a statistical mechanics model of constrained hard-core random walkers.We uncover a charge-sharpening transition that separates different scrambling phases with volume-law scaling of entanglement, distinguished by whether measurements can efficiently reveal the total charge of the system.We find that while Rényi entropies grow sub-ballistically as ffiffi t p in the absence of measurement, for even an infinitesimal rate of measurements, all average Rényi entropies grow ballistically with time ∼t.We study numerically the critical behavior of the charge-sharpening and entanglement transitions in U(1) circuits, and show that they exhibit emergent Lorentz invariance and can also be diagnosed using scalable local ancilla probes.Our statistical mechanical mapping technique readily generalizes to arbitrary Abelian groups, and offers a general framework for studying dissipatively stabilized symmetry-breaking and topological orders.
The entanglement entropy measures quantum correlations and it can be seen as the uncertainty on a quantum state. In one spatial dimension, the entanglement entropy scales as the boundary that divides two subsystems, so an area law has been proposed. However, the entanglement entropy diverges logarithmically at conformally invariant critical points, so the area law does not hold. The purpose of the work is to find a way to get more information about a critical state. The ground state of the Heisenberg XXZ model at criticality is analyzed by means of critical Ising eigenstates. Two ways of analysis are followed: a basis made of Ising eigenstates is built up and used to represent the XXZ ground state, then the Shannon entropy in the new basis is computed; the adiabatic evolution from the Ising ground state to the XXZ ground state. The result is that the Shannon entropy in the Ising basis scales linearly with the length of the system, while a phase transition is encountered during the adiabatic evolution. The conclusion is that there is no net gain in information after the procedure and possibly it is related to the fact the two systems stand in different phases.