Identifying Entanglement Phases with Bipartite Projected Ensembles

TL;DR

This work introduces bipartite projected ensembles (BPEs) and ensemble-averaged entanglement (EAE) as a novel correlation-based framework to diagnose entanglement phases in quantum many-body systems. By projecting a subsystem and studying the average entanglement between the remaining parts, the authors map volume-law, critical, and area-law regimes to long-range order, scale-invariance, and short-range order, respectively, and apply this to measurement-induced phase transitions (MIPTs) in monitored Clifford circuits. They uncover universal dynamical scaling and distinct surface critical exponents through EAE, providing a new lens to study non-equilibrium entanglement phenomena. Importantly, they propose an experimentally feasible protocol to measure EAE, with complexity scaling favorably relative to entanglement entropy measurements, enhancing prospects for quantum simulation diagnostics and broader applications to thermalization and localization physics.

Abstract

We introduce bipartite projected ensembles (BPEs) for quantum many-body wave functions, which consist of pure states supported on two local subsystems, with each state associated with the outcome of a projective measurement of the complementary subsystem in a fixed local basis. We demonstrate that the corresponding ensemble-averaged entanglements (EAEs) between two subsystems can effectively identify entanglement phases. In volume-law entangled states, EAE converges to a nonzero value with increasing distance between subsystems. For critical systems, EAE exhibits power-law decay, and it decays exponentially for area-law systems. Thus, entanglement phase transitions can be viewed as a disordered-ordered phase transition. We also apply BPE and EAE to measured random Clifford circuits to probe measurement-induced phase transitions. We show that EAE serves not only as a witness to phase transitions, but also unveils additional critical phenomena properties, including dynamical scaling and surface critical exponents. Our findings provide an alternative approach to diagnosing entanglement laws, thus enhancing the understanding of entanglement phase transitions. Moreover, given the accessibility of measuring EAE in quantum simulators, our results hold promise for impacting quantum simulations.

Figures (7)

• Figure 1: Setup. (a) The diagram of BPE. For a many-qubit system with a pure state$\ket{\Psi}$, the subsystem $R$ is measured in a fixed local basis. Then, the remaining unmeasured subsystems $A$ and $B$ are in a pure state, which depends on the measurement outcome on $R$. (b) The scaling of EAE for different entanglement laws. (c) The structure of the unitary-measurement hybrid circuit. The local two-qubit gates are drawn from the uniformly sampled Clifford group, and the random projected measurement is onto the $z$-component with a probability $p$.
• Figure 2: The results of EAE for random Clifford circuits. The dynamics of $\bar{E}(r)$ for (a) $p=0.05$, (b) $p=0.16$, and (c) $p=0.25$. The system size is $L=256$. The scaling of $\bar{E}(r)$ at $t=2L$ for (d) $p=0.05$, (e) $p=0.16$, and (f) $p=0.25$. The black dashed line in (e) is for the fit: $\bar{E}(r)\sim r^{-\eta}$, with $\eta\approx 0.71$. The sampling times are 10,000 for $L=32,64,128$, and 5,000 for $L=256$.
• Figure 3: The integrated EAE $\varrho$ versus $p$ for different system sizes. The inset is a collapse of the data, with the critical point $p_c \approx 0.16$ and exponent $\nu\approx1.24$. We choose periodic boundary conditions. The sampling times are 10,000 for $L=32,64,128$, and 5,000 for $L=256$.
• Figure 4: Critical dynamic scaling. (a) The dynamics of $\varrho^{(k)}$ at the critical point with $L=256$. The black dashed lines are linear fits. (b) Scaling of $\varrho^{(k)}$ versus $k$. The black dashed line is a linear fit with $\theta \approx0.38$ and $z \approx1.01$. (c) The dynamics of $\varrho^{(1)}$ in the disentangling phase for different $p$ with a fixed system size $L=256$. (d) Finite-size scaling of $\varrho^{(1)}(t)$ at the critical point.
• Figure 5: Surface critical phenomena. (a) The scaling of EAEs between an edge site and a bulk edge. (b) The scaling of EAEs between two edge sites. The black dashed lines are linear fits, with $\eta_\perp\approx1.02$ and $\eta_\parallel \approx1.34$.