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arXiv:2307.11609 · Physical Review Letters

Persistent Ballistic Entanglement Spreading with Optimal Control in Quantum Spin Chains

Ying Lu, Pei Shi, Xiaohan Wang, Jie Hu, Shi-Ju Ran

Published Aug 13, 2024• 4 citations
arXivHTMLPDF

Abstract

Entanglement propagation provides a key routine to understand quantum many-body dynamics in and out of equilibrium. Entanglement entropy (EE) usually approaches to a subsaturation known as the Page value S[over ˜]_{P}=S[over ˜]-dS (with S[over ˜] the maximum of EE and dS the Page correction) in, e.g., the random unitary evolutions. The ballistic spreading of EE usually appears in the early time and will be deviated far before the Page value is reached. In this work, we uncover that the magnetic field that maximizes the EE robustly induces persistent ballistic spreading of entanglement in quantum spin chains. The linear growth of EE is demonstrated to persist until the maximal S[over ˜] (along with a flat entanglement spectrum) is reached. The robustness of ballistic spreading and the enhancement of EE under such an optimal control are demonstrated, considering particularly perturbing the initial state by random pure states (RPSs). These are argued as the results from the endomorphism of the time evolution under such an entanglement-enhancing optimal control for the RPSs.

DOI
10.1103/physrevlett.133.070402

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Entanglement propagation provides a key routine to understand quantum many-body dynamics in and out of equilibrium. Entanglement entropy (EE) usually approaches to a subsaturation known as the Page value S[over ˜]_{P}=S[over ˜]-dS (with S[over ˜] the maximum of EE and dS the Page correction) in, e.g., the random unitary evolutions. The ballistic spreading of EE usually appears in the early time and will be deviated far before the Page value is reached. In this work, we uncover that the magnetic field that maximizes the EE robustly induces persistent ballistic spreading of entanglement in quantum spin chains. The linear growth of EE is demonstrated to persist until the maximal S[over ˜] (along with a flat entanglement spectrum) is reached. The robustness of ballistic spreading and the enhancement of EE under such an optimal control are demonstrated, considering particularly perturbing the initial state by random pure states (RPSs). These are argued as the results from the endomorphism of the time evolution under such an entanglement-enhancing optimal control for the RPSs.
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.
We study the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length and its complement, starting from a pure state which is not an eigenstate of the Hamiltonian. We use path integral methods of quantum field theory as well as explicit computations for the transverse Ising spin chain. In both cases, there is a maximum speed v of propagation of signals. In general the entanglement entropy increases linearly with time t up to , after which it saturates at a value proportional to , the coefficient depending on the initial state. This behaviour may be understood as a consequence of causality.
We explore the dynamics of the entanglement entropy near equilibrium in highly entangled pure states of two quantum-chaotic spin chains undergoing unitary time evolution. We examine the relaxation to equilibrium from initial states with either less or more entanglement entropy than the equilibrium value, as well as the dynamics of the spontaneous fluctuations of the entanglement that occur in equilibrium. For the spin chain with a time-independent Hamiltonian and thus an extensive conserved energy, we find slow relaxation of the entanglement entropy near equilibration. Such slow relaxation is absent in a Floquet spin chain with a Hamiltonian that is periodic in time and thus has no local conservation law. Therefore, we argue that slow diffusive energy transport is responsible for the slow relaxation of the entanglement entropy in the Hamiltonian system.
Generic quantum many-body systems typically show a linear growth of the entanglement entropy after a quench from a product state. While entanglement is a property of the wave function, it is generated by the unitary time-evolution operator and is therefore reflected in its increasing complexity as quantified by the operator entanglement entropy. Using numerical simulations of a static and a periodically driven quantum spin chain, we show that there is a robust correspondence between the entanglement entropy growth of typical product states and the operator entanglement entropy of the unitary evolution operator, while special product states, e.g., σz basis states, can exhibit faster entanglement production. In the presence of a disordered magnetic field in our spin chains, we show that both the wave function and operator entanglement entropies exhibit a power-law growth with the same disorder-dependent exponent and clarify the apparent discrepancy in previous results. These systems, in the absence of conserved densities, provide further evidence for slow information spreading on the ergodic side of the many-body localization transition.
We study the entanglement dynamics of quantum many-body systems and prove the following: (I) For any geometrically local Hamiltonian on a lattice, starting from a random product state the entanglement entropy is bounded away from the maximum entropy at all times with high probability. (II) In a spin-glass model with random all-to-all interactions, starting from any product state the average entanglement entropy is bounded away from the maximum entropy at all times. We also extend these results to any unitary evolution with charge conservation and to the Sachdev-Ye-Kitaev model. Our results highlight the difference between the entanglement generated by (chaotic) Hamiltonian dynamics and that of random states, for the latter is nearly maximal.
The non-equilibrium dynamics of disordered many-body quantum systems after a global quantum quench unveils important insights about the competition between interactions and disorder, yielding in particular an insightful perspective on many body localization (MBL). Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on the quantum quench dynamics have so far not been investigated. In this letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power ααα. Using a dynamical version of the strong disorder renormalization group (SDRG) we find for α>αcα>α_cα>αc​ that the entanglement entropy grows logarithmically with time and becomes smaller with larger ααα as S(t)=Spln⁡(t)/(2α)S(t) = S_p \ln(t)/(2α)S(t)=Sp​ln(t)/(2α). Here, Sp=2ln⁡2−1S_p= 2 \ln2 -1Sp​=2ln2−1. We use numerical exact diagonalization (ED) simulations to verify our results for system sizes up to $ N\sim 16$ spins, yielding good agreement for sufficiently large α>αc≈1.8α> α_c \approx 1.8α>αc​≈1.8. For α<αcα<α_cα<αc​, we find that the entanglement entropy grows as a power-law with time, S(t)∼tγ(α)S(t)\sim t^{γ(α)}S(t)∼tγ(α) with 0<γ(α)<10<γ(α)<10<γ(α)<1 a decaying function of the interaction exponent ααα.