Entanglement Spreading Research Articles

Spread of entanglement and causality

We investigate causality constraints on the time evolution of entanglement entropy after a global quench in relativistic theories. We first provide a general proof that the so-called tsunami velocity is bounded by the speed of light. We then generalize the free particle streaming model of arXiv:cond-mat/0503393 to general dimensions and to an arbitrary entanglement pattern of the initial state. In more than two spacetime dimensions the spread of entanglement in these models is highly sensitive to the initial entanglement pattern, but we are able to prove an upper bound on the normalized rate of growth of entanglement entropy, and hence the tsunami velocity. The bound is smaller than what one gets for quenches in holographic theories, which highlights the importance of interactions in the spread of entanglement in many-body systems. We propose an interacting model which we believe provides an upper bound on the spread of entanglement for interacting relativistic theories. In two spacetime dimensions with multiple intervals, this model and its variations are able to reproduce intricate results exhibited by holographic theories for a significant part of the parameter space. For higher dimensions, the model bounds the tsunami velocity at the speed of light. Finally, we construct a geometric model for entanglement propagation based on a tensor network construction for global quenches.

Real-Time Dynamics in U(1) Lattice Gauge Theories with Tensor Networks

Tensor network algorithms provide a suitable route for tackling real-time dependent problems in lattice gauge theories, enabling the investigation of out-of-equilibrium dynamics. We analyze a U(1) lattice gauge theory in (1+1) dimensions in the presence of dynamical matter for different mass and electric field couplings, a theory akin to quantum-electrodynamics in one-dimension, which displays string-breaking: the confining string between charges can spontaneously break during quench experiments, giving rise to charge-anticharge pairs according to the Schwinger mechanism. We study the real-time spreading of excitations in the system by means of electric field and particle fluctuations: we determine a dynamical state diagram for string breaking and quantitatively evaluate the time-scales for mass production. We also show that the time evolution of the quantum correlations can be detected via bipartite von Neumann entropies, thus demonstrating that the Schwinger mechanism is tightly linked to entanglement spreading. To present the variety of possible applications of this simulation platform, we show how one could follow the real-time scattering processes between mesons and the creation of entanglement during scattering processes. Finally, we test the quality of quantum simulations of these dynamics, quantifying the role of possible imperfections in cold atoms, trapped ions, and superconducting circuit systems. Our results demonstrate how entanglement properties can be used to deepen our understanding of basic phenomena in the real-time dynamics of gauge theories such as string breaking and collisions.

Magnetic entanglement in spin-1/2 XY chains

In the study of entanglement in a spin chain, people often consider the nearest-neighbor spins. The motivation is the prevailing role of the short range interactions in creating quantum correlation between the 1st neighbor (1N) spins. Here, we address the same question between farther neighbor spins. We consider the one-dimensional (1D) spin-1/2 XY model in a magnetic field. Using the fermionization approach, we diagonalize the Hamiltonian of the system. Then, we provide the analytical results for entanglement between the 2nd, 3rd and 4th neighbor (denoted as 2N, 3N, and 4N respectively) spins. We find a magnetic entanglement that starts from a critical entangled-field (hcE) at zero temperature. The critical entangled-field depends on the distance between the spins. In addition to the analytical results, the mentioned phenomenon is confirmed by the numerical Lanczos calculations. By adding the temperature to the model, the magnetic entanglement remains stable up to a critical temperature, Tc. Our results show that entanglement spreads step by step to farther neighbors in the spin chain by reducing temperature. At first, the 1N spins are entangled and then further neighbors will be entangled respectively. Tc depends on the value of the magnetic field and will be maximized at the quantum critical field.

Spreading of entanglement and steering along small Bose-Hubbard chains

We investigate how entanglement spreads along small Bose-Hubbard chains, with only the first well initially occupied by a mesoscopic number of atoms, as the number of sites increases. For two- and three-well chains in the non-interacting case, we are able to obtain analytical solutions and show that the presence of entanglement depends on having a sub-Poissonian state of the atoms in the first well. In these cases, the correlations we calculate are completely periodic. Restoring the collisional interactions or moving to a four-well chain necessitates a numerical treatment, for which we use the fully quantum positive-P representation. We examine two different correlations and find that adding collisional interactions destroys the periodicity of the correlations and causes them to degrade with time. This happens well before there is a noticeable effect on the periodicity of the solutions for the number of atoms in each well.

Theory of the Many-Body Localization Transition in One-Dimensional Systems

We formulate a theory of the many-body localization transition based on a novel real space renormalization group (RG) approach. The results of this theory are corroborated and intuitively explained with a phenomenological effective description of the critical point and of the "badly conducting" state found near the critical point on the delocalized side. The theory leads to the following sharp predictions: (i) The delocalized state established near the transition is a Griffiths phase, which exhibits sub-diffusive transport of conserved quantities and sub-ballistic spreading of entanglement. The anomalous diffusion exponent $\alpha < 1/2$ vanishes continuously at the critical point. The system does thermalize in this Griffiths phase. (ii) The many-body localization transition is controlled by a new kind of infinite randomness RG fixed point, where the broadly distributed scaling variable is closely related to the eigenstate entanglement entropy. Dynamically, the entanglement grows as $\sim\log t$ at the critical point, as it also does in the localized phase. (iii) In the vicinity of the critical point the ratio of the entanglement entropy to the thermal entropy, and its variance (and in fact all moments) are scaling functions of $L/\xi$, where $L$ is the length of the system and $\xi$ is the correlation length, which has a power-law divergence at the critical point.

Entanglement scrambling in 2d conformal field theory

We investigate how entanglement spreads in time-dependent states of a 1+1 dimensional conformal field theory (CFT). The results depend qualitatively on the value of the central charge. In rational CFTs, which have central charge below a critical value, entanglement entropy behaves as if correlations were carried by free quasiparticles. This leads to long-term memory effects, such as spikes in the mutual information of widely separated regions at late times. When the central charge is above the critical value, the quasiparticle picture fails. Assuming no extended symmetry algebra, any theory with $c>1$ has diminished memory effects compared to the rational models. In holographic CFTs, with $c \gg 1$, these memory effects are eliminated altogether at strong coupling, but reappear after the scrambling time $t \gtrsim \beta \log c$ at weak coupling.

Detecting two-site spin-entanglement in many-body systems with local particle-number fluctuations

We derive experimentally measurable lower bounds for the two-site entanglement of the spin-degrees of freedom of many-body systems with local particle-number fluctuations. Our method aims at enabling the spatially resolved detection of spin-entanglement in Hubbard systems using high-resolution imaging in optical lattices. A possible application is the observation of entanglement generation and spreading during spin impurity dynamics, for which we provide numerical simulations. More generally, the scheme can simplify the entanglement detection in ion chains, Rydberg atoms, or similar atomic systems.

Entanglement negativity after a global quantum quench

We study the time evolution of the logarithmic negativity after a global quantum quench. In a 1 + 1-dimensional conformal invariant field theory, we consider the negativity between two intervals which can be either adjacent or disjoint. We show that the negativity follows the quasi-particle interpretation for the spreading of entanglement. We check and generalise our findings with a systematic analysis of the negativity after a quantum quench in the harmonic chain, highlighting two peculiar lattice effects: the late birth and the sudden death of entanglement.

Phenomenology of fully many-body-localized systems

We consider fully many-body-localized systems, i.e., isolated quantum systems where all the many-body eigenstates of the Hamiltonian are localized. We define a sense in which such systems are integrable, with localized conserved operators. These localized operators are interacting pseudospins, and the Hamiltonian is such that unitary time evolution produces dephasing but not ``flips'' of these pseudospins. As a result, an initial quantum state of a pseudospin can in principle be recovered via (pseudospin) echo procedures. We discuss how the exponentially decaying interactions between pseudospins lead to logarithmic-in-time spreading of entanglement starting from nonentangled initial states. These systems exhibit multiple different length scales that can be defined from exponential functions of distance; we suggest that some of these decay lengths diverge at the phase transition out of the fully many-body-localized phase while others remain finite.

Entanglement spreading after a geometric quench in quantum spin chains

We investigate the entanglement spreading in the anisotropic spin-1/2 Heisenberg (XXZ) chain after a geometric quench. This corresponds to a sudden change of the geometry of the chain or, in the equivalent language of interacting fermions confined in a box trap, to a sudden increase of the trap size. The entanglement dynamics after the quench is associated with the ballistic propagation of a magnetization wavefront. At the free fermion point (XX chain), the von-Neumann entropy S_A exhibits several intriguing dynamical regimes. Specifically, at short times a logarithmic increase is observed, similar to local quenches. This is accurately described by an analytic formula that we derive from heuristic arguments. At intermediate times partial revivals of the short-time dynamics are superposed with a power-law increase S_A t^\alpha, with \alpha<1. Finally, at very long times a steady state develops with constant entanglement entropy. As expected, since the model is integrable, we find that the steady state is non thermal, although it exhibits extensive entanglement entropy. We also investigate the entanglement dynamics after the quench from a finite to the infinite chain (sudden expansion). While at long times the entanglement vanishes, we demonstrate that its relaxation dynamics exhibits a number of scaling properties. Finally, we discuss the short-time entanglement dynamics in the XXZ chain in the gapless phase. The same formula that describes the time dependence for the XX chain remains valid in the whole gapless phase.

Entanglement spreading in a many-body localized system

Motivated by the findings of logarithmic spreading of entanglement in a many-body localized system, we identify and untangle two factors contributing to the spreading of entanglement in the fully many-body localized phase, where all many-body eigenstates are localized. Performing full diagonalizations of an $XXZ$ spin model with random longitudinal fields, we demonstrate a linear dependence of the spreading rate on the decay length $(\ensuremath{\xi})$ of the effective interaction between localized pseudospins (l-bits), which depends on the disorder strength, and on the final value of entanglement per spin $({s}_{\ensuremath{\infty}})$, which primarily depends on the initial state. The entanglement entropy thus grows with time as $\ensuremath{\sim}\ensuremath{\xi}\ifmmode\times\else\texttimes\fi{}{s}_{\ensuremath{\infty}}logt$, providing support for the phenomenology of many-body localized systems proposed by Huse and Oganesyan.

Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System

We study the time evolution of the entanglement entropy of a one-dimensional nonintegrable spin chain, starting from random nonentangled initial pure states. We use exact diagonalization of a nonintegrable quantum Ising chain with transverse and longitudinal fields to obtain the exact quantum dynamics. We show that the entanglement entropy increases linearly with time before finite-size saturation begins, demonstrating a ballistic spreading of the entanglement, while the energy transport in the same system is diffusive. Thus, we explicitly demonstrate that the spreading of entanglement is much faster than the energy diffusion in this nonintegrable system.

Spreading of correlations and entanglement after a quench in the one-dimensionalBose–Hubbard model

We investigate the spreading of information in a one-dimensional Bose–Hubbard systemafter a sudden parameter change. In particular, we study the time evolution of correlationsand entanglement following a quench. The investigated quantities show a light-cone-likeevolution, i.e. the spreading with a finite velocity. We discuss the relation of this velocity toother characteristic velocities of the system, like the sound velocity. The entanglement isinvestigated using two different measures, the von Neumann entropy and mutualinformation. Whereas the von Neumann entropy grows rapidly with time the mutualinformation between two small subsystems can decrease as well after an initial increase.Additionally we show that the static von Neumann entropy characterizes the location ofthe quantum phase transition.

Entanglement invariant for the double Jaynes-Cummings model

We study entanglement dynamics between four qubits interacting through two isolated Jaynes-Cummings Hamiltonians, via an entanglement measure based on the wedge product. We compare the results with similar results obtained using bipartite concurrence resulting in what is referred to as ``entanglement sudden death.'' We find a natural entanglement invariant under evolution, demonstrating that entanglement spreads out over all of the system's degrees of freedom that become entangled through the interaction. We also provide an analysis of why certain initial states lose all their entanglement in a finite time, although their excitation and coherence vanish only asymptotically with time.

Relativistic effects and rigorous limits for discrete- and continuous-time quantum walks

The mathematical relationship between the discrete-time and continuous-time quantum walks and the one-dimensional Dirac equation is explored by studying a class of solutions for each, expressed in terms of the generalized, regular, and modified Bessel functions, respectively. Rigorous limits connecting these solutions are established. In addition, new analytical and numerical results are presented for quantum walks and the Dirac equation, including entanglement, relativistic localization and wave packet spreading, and normal and anomalous Zitterbewegung.

The emergence of typical entanglement in two-party random processes

We investigate the entanglement within a system undergoing a random, local process. We find that there is initially a phase of very fast generation and spread of entanglement. At the end of this phase the entanglement is typically maximal. In Oliveira et al (2007 Phys. Rev. Lett. 98 130502) we proved that the maximal entanglement is reached to a fixed arbitrary accuracy within O(N3) steps, where N is the total number of qubits. Here we provide a detailed and more pedagogical proof. We demonstrate that one can use the so-called stabilizer gates to simulate this process efficiently on a classical computer. Furthermore, we discuss three ways of identifying the transition from the phase of rapid spread of entanglement to the stationary phase: (i) the time when saturation of the maximal entanglement is achieved, (ii) the cutoff moment, when the entanglement probability distribution is practically stationary, and (iii) the moment block entanglement exhibits volume scaling. We furthermore investigate the mixed state and multipartite setting. Numerically, we find that the mutual information appears to behave similarly to the quantum correlations and that there is a well-behaved phase-space flow of entanglement properties towards an equilibrium. We describe how the emergence of typical entanglement can be used to create a much simpler tripartite entanglement description. The results form a bridge between certain abstract results concerning typical (also known as generic) entanglement relative to an unbiased distribution on pure states and the more physical picture of distributions emerging from random local interactions.

Superballistic diffusion of entanglement in disordered spin chains

We study the dynamics of a single excitation in an infinite XXZ spin chain, which is launched from the origin. We study the time evolution of the spread of entanglement in the spin chain and obtain an expression for the second order spatial moment of concurrence, about the origin, for both ordered and disordered chains. In this way, we show that a finite central disordered region can lead to sustained superballistic growth in the second order spatial moment of entanglement within the chain.