Models for non-unitary quantum dynamics, such as quantum circuits that include projective measurements, have been shown to exhibit rich quantum critical behavior. There are many complementary perspectives on this behavior. For example, there is a known correspondence between d-dimensional local non-unitary quantum circuits and tensor networks on a D=(d+1)-dimensional lattice. Here, we show that in the case of systems of non-interacting fermions, there is furthermore a full correspondence between non-unitary circuits in d spatial dimensions and unitary non-interacting fermion problems with static Hermitian Hamiltonians in D=(d+1) spatial dimensions. This provides a powerful new perspective for understanding entanglement phases and critical behavior exhibited by non-interacting circuits. Classifying the symmetries of the corresponding non-interacting Hamiltonian, we show that a large class of random circuits, including the most generic circuits with randomness in space and time, are in correspondence with Hamiltonians with static spatial disorder in the ten Altland-Zirnbauer symmetry classes. We find the criticality that is known to occur in all of these classes to be the origin of the critical entanglement properties of the corresponding random non-unitary circuit. To exemplify this, we numerically study the quantum states at the boundary of Haar-random Gaussian fermionic tensor networks of dimension D=2 and D=3. We show that the most general such tensor network ensemble corresponds to a unitary problem of non-interacting fermions with static disorder in Altland-Zirnbauer symmetry class DIII, which for both D=2 and D=3 is known to exhibit a stable critical metallic phase. Tensor networks and corresponding random non-unitary circuits in the other nine Altland-Zirnbauer symmetry classes can be obtained from the DIII case by implementing Clifford algebra extensions for classifying spaces.
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.
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…
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 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 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 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…
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…
We explore the dynamical phases of unitary Clifford circuits with variable-range interactions, coupled to a monitoring environment. We investigate two classes of models, distinguished by the action of the unitary gates, which either are organized in clusters of finite-range two-body gates, or are pair-wise interactions randomly distributed throughout the system with a power-law distribution. We find the range of the interactions plays a key role in characterizing both phases and their measurement-induced transitions. For the cluster unitary gates we find a transition between a phase with volum…
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 an…
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, …
We investigate the critical behavior of the entanglement transition induced by projective measurements in (Haar) random unitary quantum circuits. Using a replica approach, we map the calculation of the entanglement entropies in such circuits onto a two-dimensional statistical mechanics model. In this language, the area- to volume-law entanglement transition can be interpreted as an ordering transition in the statistical mechanics model. We derive the general scaling properties of the entanglement entropies and mutual information near the transition using conformal invariance. We analyze in det…
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…
Starting from a state of low quantum entanglement, local unitary time evolution increases the entanglement of a quantum many-body system. In contrast, local projective measurements disentangle degrees of freedom and decrease entanglement. We study the interplay of these competing tendencies by considering time evolution combining both unitary and projective dynamics. We begin by constructing a toy model of Bell pair dynamics which demonstrates that measurements can keep a system in a state of low (i.e. area law) entanglement, in contrast with the volume law entanglement produced by generic pur…
We study the growth of genuine multipartite entanglement in random quantum circuit models, which include random unitary circuit models and the random Clifford circuit. We find that for the random Clifford circuit, the growth of multipartite entanglement remains slower in comparison to the random unitary case. However, the final saturation value of multipartite entanglement is almost the same in both cases. The behavior is then compared to the genuine multipartite entanglement obtained in random matrix product states with a moderately high bond dimension. We then relate the behavior of multipar…
We construct a random unitary Gaussian circuit for continuous-variable (CV) systems subject to Gaussian measurements. We show that when the measurement rate is nonzero, the steady state entanglement entropy saturates to an area-law scaling. This is different from a many-body qubit system, where a generic entanglement transition is widely expected. Due to the unbounded local Hilbert space, the time scale to destroy entanglement is always much shorter than the one to build it, while a balance could be achieved for a finite local Hilbert space. By the same reasoning, the absence of transition sho…
Entanglement transitions in quantum dynamics present a novel class of phase transitions in non-equilibrium systems. When a many-body quantum system undergoes unitary evolution interspersed with monitored random measurements, the steady-state can exhibit a phase transition between volume and area-law entanglement. There is a correspondence between measurement-induced transitions in non-unitary quantum circuits in $d$ spatial dimensions and classical statistical mechanical models in $d+1$ dimensions. In certain limits these models map to percolation, but there is analytical and numerical evidenc…
The generating functions for the gauge theory observables are often represented in terms of the unitary matrix integrals. In this work, the perturbative and non-perturbative aspects of the generic multi-critical unitary matrix models are studied by adopting the integrable operator formalism, and the multi-critical generalization of the Tracy--Widom distribution in the context of random partitions. We obtain the universal results for the multi-critical model in the weak and strong coupling phases. The free energy of the instanton sector in the weak coupling regime, and the genus expansion of th…
In quantum statistical mechanics, finite-temperature phase transitions are typically governed by classical field theories. In this context, the role of quantum correlations is unclear: recent contributions have shown how entanglement is typically very short-ranged, and thus uninformative about long-ranged critical correlations. In this work, we show the existence of finite-temperature phase transitions where a broader form of quantum correlation than entanglement, the entropic quantum discord, can display genuine signatures of critical behavior. We consider integrable bosonic field theories in…
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…
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…
Time evolution of quantum many-body systems typically leads to a state with maximal entanglement allowed by symmetries. Two distinct routes to impede entanglement growth are inducing localization via spatial disorder, or subjecting the system to non-unitary evolution, e.g., via projective measurements. Here we employ the idea of space-time rotation of a circuit to explore the relation between systems that fall into these two classes. In particular, by space-time rotating unitary Floquet circuits that display a localization transition, we construct non-unitary circuits that display a rich varie…
We employ unsupervised learning tools to identify different phases and their transition in quantum systems subject to the combined action of unitary evolution and stochastic measurements. Specifically, we consider principal component analysis and intrinsic dimension estimation to reveal a measurement-induced structural transition in the data space. We test our approach on a 1+1D stabilizer circuit and find the quantities of interest furnish novel order parameters defined directly in the raw data space. Our results provide a first use of unsupervised tools in dynamical quantum phase transitions…
We investigate the dynamics of two-dimensional quantum spin systems under the combined effect of random unitary gates and local projective measurements. When considering steady states, a measurement-induced transition occurs between two distinct dynamical phases, one characterized by a volume-law scaling of entanglement entropy, the other by an area-law. Employing stabilizer states and Clifford random unitary gates, we numerically investigate square lattices of linear dimension up to $L=48$ for two distinct measurement protocols. For both protocols, we observe a transition point where the domi…
We numerically explore the many body localization (MBL) transition through the lens of the {\it entanglement spectrum}. While a direct transition from localization to thermalization is believed to obtain in the thermodynamic limit (the exact details of which remain an open problem), in finite system sizes there exists an intermediate `quantum critical' regime. Previous numerical investigations have explored the crossover from thermalization to criticality, and have used this to place a numerical {\it lower} bound on the critical disorder strength for MBL. A careful analysis of the {\it high en…
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…
We consider a dynamic protocol for quantum many-body systems, which enables to study the interplay between unitary Hamiltonian driving and random local projective measurements. While the unitary dynamics tends to increase entanglement, local measurements tend to disentangle, thus favoring decoherence. Close to a quantum transition where the system develops critical correlations with diverging length scales, the competition of the two drivings is analyzed within a dynamic scaling framework, allowing us to identify a regime (dynamic scaling limit) where the two mechanisms develop a nontrivial in…
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…
We study the critical behavior and the ground-state entanglement of a large class of $\mathrm{su}(1|1)$ supersymmetric spin chains with a general (not necessarily monotonic) dispersion relation. We show that this class includes several relevant models, with both short- and long-range interactions of a simple form. We determine the low temperature behavior of the free energy per spin, and deduce that the models considered have a critical phase in the universality class of a $(1+1)$-dimensional conformal field theory (CFT), whose central charge coincides with the number of connected components o…
Characterizing how entanglement grows with time in a many-body system, for example after a quantum quench, is a key problem in non-equilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time--dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the `entanglement tsunami' in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow acco…
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…
Local unitary operations allow for a unifying approach to the quantification of quantum correlations among the constituents of a bipartite quantum system. For pure states, the distance between a given state and its image under least-perturbing local unitary operations is a bona fide measure of quantum entanglement, the so-called entanglement of response, which can be extended to mixed states via the convex roof construction. On the other hand, when defined directly on mixed states perturbed by local unitary operations, such a distance turns out to be a bona fide measure of quantum correlations…
We study generalizations of the Gross--Witten--Wadia unitary matrix model for the special orthogonal and symplectic groups. We show using a standard Coulomb gas treatment -- employing a path integral formalism for the ungapped phase and resolvent techniques for the gapped phase with one coupling constant -- that in the large $N$ limit, the free energy normalized modulo the square of the gauge group rank is twice the value for the unitary case. Using generalized Cauchy identities for character polynomials, we then demonstrate the universality of this phase transition for an arbitrary number of …
A formidable perspective in understanding quantum criticality of a given many-body system is through its entanglement contents. Until now, most progress are only limited to the disorder-free case. Here, we develop an efficient scheme to compute the entanglement entropy of $(2+1)$-dimensional quantum critical points with randomness, from a conceptually novel angle where the quenched disorder can be considered as dimensionally reducible interactions. As a concrete example, we reveal novel entanglement signatures of $(2+1)$-dimensional Dirac fermion exposed to a random magnetic field, which hosts…
A global quantum quench can be modeled by a quantum circuit with local unitary gates. In general, entanglement grows linearly at a rate given by entanglement velocity, which is upper bounded by the growth of the light cone. We show that the unitary interactions achieving the maximal rate must remain unitary if we exchange the space and time directions -- a property known as dual unitarity. Our results are robust: approximate maximal entanglement velocity also implies approximate dual unitarity. We further show that maximal entanglement velocity is always accompanied by a specific dynamical pat…
The entanglement in operator space is a well established measure for the complexity of the quantum many-body dynamics. In particular, that of local operators has recently been proposed as dynamical chaos indicator, i.e. as a quantity able to discriminate between quantum systems with integrable and chaotic dynamics. For chaotic systems the local-operator entanglement is expected to grow linearly in time, while it is expected to grow at most logarithmically in the integrable case. Here we study local-operator entanglement in dual-unitary quantum circuits, a class of "statistically solvable" quan…
We map the dynamics of entanglement in random unitary circuits, with finite on-site Hilbert space dimension $q$, to an effective classical statistical mechanics, and develop general diagrammatic tools for calculations in random unitary circuits. We demonstrate explicitly the emergence of a `minimal membrane' governing entanglement growth, which in 1+1D is a directed random walk in spacetime (or a variant thereof). Using the replica trick to handle the logarithm in the definition of the $n$th Rényi entropy $S_n$, we map the calculation of the entanglement after a quench to a problem of interact…
