We describe conditions for generating entanglement between two regions at the optimal rate in a class of one-dimensional quantum circuits with Floquet dynamics. The optimal value follows from subadditivity and Araki-Lieb inequalities. A quantum circuit composed of parallel SWAP gates that act periodically on entangled pairs is a simple system that saturates the bound. We show that any other system that entangles at this maximal rate must act as a generalized SWAP gate dynamics on the relevant states of the Hilbert space. We further discuss some characterizations of states according to entropy generation. States with multipartite entanglement generically fail to entangle efficiently as time evolves. This suggests that chaos, which tend to produce such entanglement patterns, is expected to work against the process of spreading information efficiently. It also provides a simple intuition for why the entangling tsunami velocity must be slower than the Lieb-Robinson velocity.
We describe conditions for generating entanglement between two regions at the optimal rate in a class of one-dimensional quantum circuits with Floquet dynamics. The optimal value follows from subadditivity and Araki-Lieb inequalities. A quantum circuit composed of parallel SWAP gates that act periodically on entangled pairs is a simple system that saturates the bound. We show that any other system that entangles at this maximal rate must act as a generalized SWAP gate dynamics on the relevant states of the Hilbert space. We further discuss some characterizations of states according to entropy generation. States with multipartite entanglement generically fail to entangle efficiently as time evolves. This suggests that chaos, which tend to produce such entanglement patterns, is expected to work against the process of spreading information efficiently. It also provides a simple intuition for why the entangling tsunami velocity must be slower than the Lieb-Robinson velocity.
We solve a minimal model for an ergodic phase in a spatially extended quantum many-body system.The model consists of a chain of sites with nearest-neighbor coupling under Floquet time evolution.Quantum states at each site span a q-dimensional Hilbert space, and time evolution for a pair of sites is generated by a q 2 × q 2 random unitary matrix.The Floquet operator is specified by a quantum circuit of depth two, in which each site is coupled to its neighbor on one side during the first half of the evolution period and to its neighbor on the other side during the second half of the period.We show how dynamical behavior averaged over realizations of the random matrices can be evaluated using diagrammatic techniques and how this approach leads to exact expressions in the large-q limit.We give results for the spectral form factor, relaxation of local observables, bipartite entanglement growth, and operator spreading.
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.
Entanglement is a defining feature of many-body quantum systems and is an essential requirement for quantum computing. It is therefore useful to study physical processes which generate entanglement within a large system, as they maybe replicated for applications involving the said requirements in quantum information processing. A possible avenue to maximize entanglement generation is to rely on the phenomena of information scrambling, i.e. transport of initially localized information throughout the system. Here the rationale is that the spread of information carries with it an inherent capacity of entanglement generation. Scrambling greatly depends upon the dynamical nature of the system Hamiltonian, and the interplay between entanglement generation and information scrambling maybe investigated taking a chain of interacting spins on a one dimensional lattice. This system is analogous to an array of qubits and this relative simplicity implies that the resulting unitary dynamics can be efficiently simulated using present-day cloud based NISQ devices. In our present work, we consider such a spin model which is made up of nearest and next nearest neighbor XXZ Model, along with an introduced coupling term λ. This coupling term serves as a tuning parameter which modifies the dynamical nature of the system from the integrable to the quantum chaotic regime. In order to quantify the entanglement generated within the system we use the more general multipartite metric which computes the average entanglement across all system bipartitions to obtain a global picture of the entanglement structure within the entire system. Further to make our study initial state independent, we consider an ensemble of random initial states and evaluate the multipartite entanglement generated for each of these states under time evolution. We find clear differences in the entanglement measures as the system undergoes through different regimes. We also analyze the fluctuations present within our entanglement metric and find a distinct pattern as we head from the integrable to the chaotic regime.
The dynamics of quantum many-body systems in the chaotic regime are of particular interest due to the associated phenomena of information scrambling and entanglement generation within the system. While these systems are typically intractable using traditional numerical methods, an effective framework can be implemented based on dual-unitary circuits which have emerged as a minimal model for maximally chaotic dynamics. In this work, we investigate how individual two-body operators influence the global dynamics of circuits composed of dual-unitaries. We study their effect on entanglement generation while examining it from both bipartite and multipartite perspectives. Here we also highlight the significant role of local unitaries in the dynamics when paired with operators from the dual-unitary class, showing that systems with identical entangling power can exhibit a range of differing entanglement growth rates. Furthermore, we present calculations establishing time-step-dependent lower bounds, which depend on both the initial state and the entangling power of the constituent operators. Finally, we find that time-evolving an initial state composed of pair products generates a state with nearly maximal multipartite entanglement content, approaching the bounds established by Absolutely Maximally Entangled (AME) states.
Dual-unitary circuits are a class of locally interacting quantum many-body systems displaying unitary dynamics also when the roles of space and time are exchanged. These systems have recently emerged as a remarkable framework where certain features of many-body quantum chaos can be studied exactly. In particular, they admit a class of ``solvable'' initial states for which, in the thermodynamic limit, one can access the full nonequilibrium dynamics. This reveals a surprising property: when a dual-unitary circuit is prepared in a solvable state the quantum entanglement between two complementary spatial regions grows at the maximal speed allowed by the local structure of the evolution. Here we investigate the fate of this property when the system is prepared in a generic pair-product state. We show that in this case, the entanglement increment during a time step is submaximal for finite times, however, it approaches the maximal value in the infinite-time limit. This statement is proven rigorously for dual-unitary circuits generating high enough entanglement, while it is argued to hold for the entire class.
The characterization of ensembles of many-qubit random states and their realization via quantum circuits are crucial tasks in quantum-information theory. In this work, we study the ensembles generated by quantum circuits that randomly permute the computational basis, thus acting classically on the corresponding states. We focus on the averaged entanglement and present two main results. First, we derive generically tight upper bounds on the entanglement that can be generated by applying permutation circuits to arbitrary initial states. We show that the late-time “entanglement Page curves” are bounded in terms of the initial state participation entropies and their overlap with the “maximally antilocalized” state. Generally speaking, this result states that the quantum correlations generated by classical circuits are bounded in terms of some quantum property of the initial state (namely, the degree to which it can be written as a superposition of classical states). Second, comparing the averaged Rényi-2 entropies generated by ( 1 ) an infinitely deep random circuit of two-qubit gates and ( 2 ) global random permutations, we show that the two quantities are different for finite N but the corresponding Page curves coincide in the thermodynamic limit. We also discuss how these conclusions are modified by additional random phases or considering circuits of k -local gates with k ≥ 3 . Our results are exact and highlight the implications of classical features on entanglement generation in many-body systems.