We analyze the quantum trajectory dynamics of free fermions subject to continuous monitoring. For weak monitoring, we identify a novel dynamical regime of subextensive entanglement growth, reminiscent of a critical phase with an emergent conformal invariance. For strong monitoring, however, the dynamics favors a transition into a quantum Zeno-like area-law regime. Close to the critical point, we observe logarithmic finite size corrections, indicating a Berezinskii-Kosterlitz-Thouless mechanism underlying the transition. This uncovers an unconventional entanglement transition in an elementary, physically realistic model for weak continuous measurements. In addition, we demonstrate that the measurement aspect in the dynamics is crucial for whether or not a phase transition takes place.
It has recently been discovered that random quantum circuits provide an avenue to realize rich entanglement phase diagrams, which are hidden to standard expectation values of operators. Here we study (2+1)D random circuits with random Clifford unitary gates and measurements designed to stabilize trivial area law and topologically ordered phases. With competing single qubit Pauli-Z and toric code stabilizer measurements, in addition to random Clifford unitaries, we find a phase diagram involving a tricritical point that maps to (2+1)D percolation, a possibly stable critical phase, topologically…
Given an output wavefunction of a monitored quantum circuit consisting of both unitary gates and projective measurements, we ask whether two complementary subsystems are entangled or not. For Clifford circuits, we find that this question can be mapped to a certain classical error-correction problem where various entanglement measures can be explicitly computed from the recoverability. The dual classical code is constructed from spacetime patterns of out-of-time ordered correlation functions among local operators and measured Pauli operators in the past, suggesting that the volume-law entanglem…
We define an ensemble of random Clifford quantum circuits whose output state undergoes an entanglement phase transition between two volume-law phases as a function of measurement rate. Our setup maps exactly the output state to the ground space of a spin glass model. We identify the entanglement phases using an order parameter that is accessible on a quantum chip. We locate the transition point and evaluate a critical exponent, revealing spin glass criticality. Our work establishes an exact statistical mechanics theory of an entanglement phase transition.
Measurement-induced criticality for non-Abelian anyons: The monitored golden chain
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Institute for Theoretical Physics, University of Cologne（科隆大学理论物理研究所）；Department of Theoretical Physics, University of Geneva（日内瓦大学理论物理系）；Geneva Quantum Center, University of Geneva（日内瓦大学日内瓦量子中心）；Department of Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校物理系）
Monitored quantum dynamics reveal quantum state trajectories which exhibit a rich phenomenology of entanglement structures, including a transition from a weakly-monitored volume law entangled phase to a strongly-monitored area law phase. For one-dimensional hybrid circuits with both random unitary dynamics and interspersed measurements, we combine analytic mappings to an effective statistical mechanics model with extensive numerical simulations on hybrid Clifford circuits to demonstrate that the universal entanglement properties of the volume law phase can be quantitatively described by a fluc…
Many elusive quantum phenomena emerge from the interaction of a quantum system with its classical environment. Quantum simulators enable us to program this interaction by using measurement operations. Measurements generally remove part of the quantum entanglement built between the qubits inside a simulator. While in simple cases entanglement may disappear at a constant rate as we measure qubits one by one, the evolution of entanglement under measurements for a given class of quantum states is generally unknown. Here, we show that consecutive measurements of the qubits in a quantum simulator ca…
We investigate entanglement dynamics in continuously monitored open quantum systems featuring current-carrying non-equilibrium states. We focus on a prototypical one-dimensional model of boundary-driven non-interacting fermions with monitoring of the local density, whose average Lindblad dynamics features a well-studied ballistic to diffusive crossover in transport. Here we analyze the dynamics of the fermionic negativity, mutual information, and purity along different quantum trajectories. We show that monitoring this boundary-driven system enhances its entanglement negativity at long times, …
The evolution of entanglement entropy in quantum circuits composed of Haar-random gates and projective measurements shows versatile behavior, with connections to phase transitions and complexity theory. We reformulate the problem in terms of a classical Markov process for the dynamics of bipartition purities and establish a probabilistic cellular-automaton algorithm to compute entanglement entropy in monitored random circuits on arbitrary graphs. In one dimension, we further relate the evolution of the entropy to a simple classical spin model that naturally generalizes a two-dimensional lattic…
Fractal structure of multipartite entanglement in monitored quantum circuits
Smalley-Curl Institute, Rice University（莱斯大学斯马利-柯尔研究所）；Department of Physics and Astronomy, Rice University（莱斯大学物理与天文系）；Laboratory of Atomic and Solid State Physics, Cornell University（康奈尔大学原子与固态物理实验室）
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 non-unitary 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, disti…
We derive exact expressions for the local entanglement entropy E in the ground state of the one-dimensional Hubbard model at a quantum phase transition driven by a change in magnetic field h or chemical potential u. The leading divergences of dE/dh and dE/du are shown to be directly related to those of the zero-temperature spin and charge susceptibilities. Logarithmic corrections to scaling signal a change in the number of local states accessible to the system as it undergoes the transition.
We study entanglement dynamics in hybrid $\mathbb{Z}_2$-symmetric quantum automaton circuits subject to local composite measurements. We show that there exists an entanglement phase transition from a volume law phase to a critical phase by varying the measurement rate $p$. By analyzing the underlying classical bit string dynamics, we demonstrate that the critical point belongs to parity-conserving universality class. We further show that the critical phase with $p>p_c$ is related to the diffusion-annihilation process and is protected by the $\mathbb{Z}_2$-symmetric measurement. We give an inte…
We study the entanglement entropy and entanglement spectrum of the paradigmatic Bose-Hubbard model, describing strongly correlated bosons on a lattice. The use of a controlled approximation - the slave-boson approach - allows us to study entanglement in all regimes of the model (and, most importantly, across its superfluid/Mott-insulator transition) at a minimal cost. We find that the area-law scaling of entanglement -- verified in all the phases -- exhibits a sharp singularity at the transition. The singularity is greatly enhanced when the transition is crossed at fixed, integer filling, due …
We study a free fermion model where two sets of non-commuting non-projective measurements stabilize area-law entanglement scaling phases of distinct topological order. We show the presence of a topological phase transition that is of a different universality class than that observed in stroboscopic projective circuits. In the presence of unitary dynamics, the two topologically distinct phases are separated by a region with sub-volume scaling of the entanglement entropy. We find that this entanglement transition is well identified by a combination of the bipartite entanglement entropy and the t…
Generic many-body systems coupled to an environment lose their quantum entanglement due to decoherence and evolve to a mixed state with only classical correlations. Here, we show that measurements can stabilize quantum entanglement within open quantum systems. Specifically, in random unitary circuits with dephasing at the boundary, we find both numerically and analytically that projective measurements performed at a small nonvanishing rate results in a steady state with an $L^{1/3}$ power-law scaling entanglement negativity within the system. Using an analytical mapping to a statistical mechan…
We propose an entanglement mean field theory inspired approach for dealing with interacting classical many-body systems. It involves a coarse-graining technique that terminates a step before the mean field theory: While mean field theory deals with only single-body physical parameters, the entanglement mean field theory deals with single- as well as two-body ones. We improve the theory to a cluster entanglement mean field, that deals with a fundamental unit of the lattice of the many-body system. We apply these methods to interacting Ising spin systems in several lattice geometries and dimensi…
We study a (1+1)-dimensional quantum circuit consisting of Haar-random unitary gates and projective measurements that conserve a total $U(1)$ charge and thus have $U(1)$ symmetry. In addition to a measurement-induced entanglement transition between a volume-law and an area-law entangled phase, we find a phase transition between two phases characterized by bipartite charge fluctuation growing with the subsystem size or staying constant. At this charge-fluctuation transition, steady-state quantities obtained by evolving an initial state with a definitive total charge exhibit critical scaling beh…
Much has been learned about universal properties of entanglement entropies in ground states of quantum many-body lattice systems. Here we unveil universal properties of the average bipartite entanglement entropy of eigenstates of the paradigmatic quantum Ising model in one dimension. The leading term exhibits a volume-law scaling that we argue is universal for translationally invariant quadratic models. The subleading term is constant at the critical field for the quantum phase transition and vanishes otherwise (in the thermodynamic limit), i.e., the critical field can be identified from suble…
Absence of measurement- and unraveling-induced entanglement transitions in continuously monitored one-dimensional free fermions
Institute for Theoretical Physics, University of Innsbruck（因斯布鲁克大学理论物理研究所）；Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）
Super-Logarithmic Entanglement Scaling in a Monitored Superconducting Chain
Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-sen University（中山大学物理学院、广东省铁电物理与器件重点实验室、中子科学与技术中心）；Max-Planck-Institut für Quantenoptik（马克斯·普朗克量子光学研究所）
Effective Hamiltonian description on monitored Majorana chains: correlated power-law hoppings and unconventional entanglement scaling
University of Tokyo（东京大学）；RIKEN Pioneering Research Institute (PRI)（理化学研究所先锋研究本部）；University of Warwick（华威大学）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所交叉理论与数学科学中心）；RIKEN（理化学研究所）；Ritsumeikan University（立命馆大学）
We propose entanglement negativity as a fine-grained probe of measurement-induced criticality. We motivate this proposal in stabilizer states, where for two disjoint subregions, comparing their "mutual negativity" and their mutual information leads to a precise distinction between bipartite and multipartite entanglement. In a measurement-only stabilizer circuit that maps exactly to two-dimensional critical percolation, we show that the mutual information and the mutual negativity are governed by boundary conformal fields of different scaling dimensions at long distances. We then consider a cla…
Quantum circuit dynamics with local projective measurements can realize a rich spectrum of entangled states of quantum matter. Motivated by the physics of the Kitaev quantum spin liquid [1], we study quantum circuit dynamics in (2+1)-dimensions involving local projective measurements, in which the monitored trajectories realize (i) a phase with topological quantum order or (ii) a "critical" phase with a logarithmic violation of area-law-scaling of the entanglement entropy along with long range tripartite entanglement. A Majorana parton description of these dynamics, which provides an out-of-eq…
We describe an algorithm for studying the entanglement entropy and spectrum of 2D systems, as a coupled array of $N$ one dimensional chains in their continuum limit. Using the algorithm to study the quantum Ising model in 2D, (both in its disordered phase and near criticality) we confirm the existence of an area law for the entanglement entropy and show that near criticality there is an additive piece scaling as $c_{eff}\log (N)/6$ with $c_{eff} \approx 1$. \textcolor{black}{Studying the entanglement spectrum, we show that entanglement gap scaling can be used to detect the critical point of th…
Tensor-network methods are used to perform a comparative study of the two-dimensional classical Heisenberg and $\mathrm{RP}^2$ models. We demonstrate that uniform matrix product states (MPS) with explicit $\mathrm{SO}(3)$ symmetry can probe correlation lengths up to $\mathcal{O}(10^3)$ sites accurately, and we study the scaling of entanglement entropy and universal features of MPS entanglement spectra. For the Heisenberg model, we find no signs of a finite-temperature phase transition, supporting the scenario of asymptotic freedom. For the $\mathrm{RP}^2$ model we observe an abrupt onset of sc…
The study of entanglement spectra is a powerful tool to detect or elucidate universal behaviour in quantum many-body systems. We investigate the scaling of the entanglement (or Schmidt) gap $δξ$, i.e., the lowest laying gap of the entanglement spectrum, at a two-dimensional quantum critical point. We focus on the paradigmatic quantum spherical model, which exhibits a second-order transition, and is mappable to free bosons with an additional external constraint. We analytically show that the Schmidt gap vanishes at the critical point, although only logarithmically. For a system on a torus and t…
Entanglement exhibits universal behavior near the ground-state critical point where correlations are long-ranged and the thermodynamic entropy is vanishing. On the other hand, a quantum quench imparts extensive energy and results in a build-up of entropy, hence no critical behavior is expected at long times. In this work, we present a new paradigm in the quench dynamics of integrable spin chains which exhibit a ground-state order-disorder phase transition at a critical line. Specifically, we consider a quench along the critical line which displays a volume-law behavior of the entropy and expon…
