We propose a phase-space Wigner harmonics entropy measure for many-body quantum dynamical complexity. This measure, which reduces to the well-known measure of complexity in classical systems and which is valid for both pure and mixed states in single-particle and many-body systems, takes into account the combined role of chaos and entanglement in the realm of quantum mechanics. The effectiveness of the measure is illustrated in the example of the Ising chain in a homogeneous tilted magnetic field. We provide numerical evidence that the multipartite entanglement generation leads to a linear increase in entropy until saturation in both integrable and chaotic regimes, so that in both cases the number of harmonics of the Wigner function grows exponentially with time. The entropy growth rate can be used to detect quantum phase transitions. The proposed entropy measure can also distinguish between integrable and chaotic many-body dynamics by means of the size of long-term fluctuations which become smaller when quantum chaos sets in.
We propose a phase-space Wigner harmonics entropy measure for many-body quantum dynamical complexity. This measure, which reduces to the well-known measure of complexity in classical systems and which is valid for both pure and mixed states in single-particle and many-body systems, takes into account the combined role of chaos and entanglement in the realm of quantum mechanics. The effectiveness of the measure is illustrated in the example of the Ising chain in a homogeneous tilted magnetic field. We provide numerical evidence that the multipartite entanglement generation leads to a linear increase in entropy until saturation in both integrable and chaotic regimes, so that in both cases the number of harmonics of the Wigner function grows exponentially with time. The entropy growth rate can be used to detect quantum phase transitions. The proposed entropy measure can also distinguish between integrable and chaotic many-body dynamics by means of the size of long-term fluctuations which become smaller when quantum chaos sets in.
We propose the Wigner separability entropy as a measure of complexity of a quantum state. This quantity measures the number of terms that effectively contribute to the Schmidt decomposition of the Wigner function with respect to a chosen phase-space decomposition. We prove that the Wigner separability entropy is equal to the operator space entanglement entropy, measuring entanglement in the space of operators, and, for pure states, twice the entropy of entanglement. The quantum to classical correspondence between the Wigner separability entropy and the separability entropy of the classical phase-space Liouville density is illustrated by means of numerical simulations of chaotic maps. In this way, the separability entropy emerges as an extremely broad complexity quantifier in both the classical and quantum realms.
We discuss the connection between the out-of-time-ordered correlator and the number of harmonics of the phase-space Wigner distribution function.In particular, we show that both quantities grow exponentially for chaotic dynamics, with a rate determined by the largest Lyapunov exponent of the underlying classical dynamics, and algebraically-linearly or quadratically-for integrable dynamics.It is then possible to use such quantities to detect in the time domain the integrability-to-chaos crossover in many-body quantum systems.
There is a remarkable interest in the study of out-of-time ordered correlators (OTOCs) that goes from many-body theory and high-energy physics to quantum chaos. In the latter case there is a special focus on the comparison with the traditional measures of quantum complexity such as the spectral statistics. The exponential growth has been verified for many paradigmatic maps and systems. However, less is known for multipartite cases. On the other hand, the recently introduced Wigner separability entropy (WSE) and its classical counterpart provide a complexity measure that treats equally quantum and classical distributions in phase space. We compare the behavior of these measures in a system consisting of two coupled and perturbed cat maps with different dynamics: Double hyperbolic, double elliptic, and mixed. In all cases, we find that the OTOCs and the WSE have essentially the same behavior, providing a complete characterization in generic bipartite systems and at the same time revealing them as very good measures of quantum complexity for phase-space distributions. Moreover, we establish a relation between both quantities by means of a recently proven theorem linking the second Rényi entropy and OTOCs.
The entangling power and operator entanglement entropy are state-independent measures of entanglement. Their growth and saturation is examined in the time-evolution operator of quantum many-body systems that can range from the integrable to the fully chaotic. An analytically solvable integrable model of the kicked transverse-field Ising chain is shown to have ballistic growth of operator von Neumann entanglement entropy and exponentially fast saturation of the linear entropy with time. Surprisingly, a fully chaotic model with longitudinal fields turned on shares the same growth phase, and is consistent with a random matrix model that is also exactly solvable for the linear entropy entanglements. However, an examination of the entangling power shows that its largest value is significantly less than the nearly maximal value attained by the nonintegrable one. The importance of long-range spectral correlations, and not just the nearest-neighbor spacing, is pointed out in determining the growth of entanglement in nonintegrable systems. Finally, an interesting case that displays some features peculiar to both integrable and nonintegrable systems is briefly discussed.
The determination of structures that can emerge out of the interactions among the constituents of a quantum many-body system is a foundational task of condensed matter physics.One of the most advanced proposals within this paradigm is the holographic generation of spacetime in some strongly coupled chaotic systems with particular patterns of entanglement and quantum complexity, which is historically motivated by the holographic principle and by the AdS/CFT correspondence.As a part of this scenario, this thesis is dedicated to the analysis of some concepts that are relevant to such proposal, where they are separately applied to some lattice models.Concretely, the matter is divided into three main studies: first, the calculation of the time-dependent circuit complexity in the Ising model with a periodically driven transverse field, where we establish the effectiveness of this quantity for the detection of nonequilibrium quantum phase transitions.Our results provide hints for understanding how universal features out of equilibrium are captured by the complexity of quantum states.Second, the derivation of a bound on the maximal rate of entanglement entropy production for a class of one-dimensional quantum circuits with periodic dynamics.An example of a circuit that saturates the bound is composed by parallel SWAP gates acting on entangled pairs.Out of inequalities obeyed by the entropy, in addition to considerations on multipartite entanglement, we indicate that the effect of a chaotic dynamics cannot result in the increase on the rate of entanglement production.Third, the construction of a class of supersymmetric many-body systems using symmetric inverse semigroups.For particular toy models built out of this structure, we study some questions regarding supersymmetric phases, integrability, disorder, spreading of quantum information and many-body localization.Finally, besides those three works, we include an essay addressing the problem of emergent holographic spaces out of two-dimensional conformal field theories, where the mediation between the two parts is performed by means of a Riemannian structure defined in the Hilbert space of the field theory.
We study quantum dynamics through the lens of entanglement generation and characterize the underlying unitary evolution by the distinct signatures it imprints on the time-dependent entangling power. For a unitary operator, we characterize ergodicity by the equality between its long-time-averaged entangling power and the Haar-averaged linear entropy. We define mixing more stringently as the convergence of the time-dependent entangling power itself to the Haar value at long times. Within this framework, we establish the ergodic hierarchy of dynamical behavior, showing in particular that mixing implies ergodicity, whereas ergodicity does not necessarily imply mixing. As an application, we find that two-qubit unitary gates are neither ergodic nor mixing: their long-time-averaged entangling power can take only four discrete values, none of which coincides with the Haar average. We then investigate many-body dynamics using the kicked Ising chain and find that the long-time-averaged entangling power converges to the Haar value in both integrable and nonintegrable cases, indicating ergodicity. Remarkably, however, the nonintegrable chain exhibits mixing, whereas the integrable chain, despite being ergodic, is demonstrably nonmixing. We also introduce a Lyapunov-like exponent to characterize the rate at which the time-dependent entangling power approaches its saturation value. We find that this exponent increases systematically with the degree of integrability breaking in the many-body system. Our results establish entanglement generation as a useful framework for characterizing dynamical systems and reveal qualitatively different signatures of integrability beyond conventional diagnostics.