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arXiv:1702.03929 · Physical review. B./Physical review. B

Information propagation in isolated quantum systems

David J. Luitz, Yevgeny Bar Lev

Published Jul 12, 2017• 174 citations
arXivHTMLPDF

Abstract

How fast does quantum information spread in isolated quantum systems? The authors address this question using out-of-time order correlation functions and find that information spreads behind a power law front in disordered spin chains, with a characteristic exponent, which is related to the exponent of the power law growth of the entanglement entropy. The authors also observe that disorder inhibits information spreading and leads to unsaturated Lieb-Robinson bounds.

DOI
10.1103/physrevb.96.020406

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How fast does quantum information spread in isolated quantum systems? The authors address this question using out-of-time order correlation functions and find that information spreads behind a power law front in disordered spin chains, with a characteristic exponent, which is related to the exponent of the power law growth of the entanglement entropy. The authors also observe that disorder inhibits information spreading and leads to unsaturated Lieb-Robinson bounds.
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.
The Lieb-Robinson theorem states that information propagates with a finite velocity in quantum systems on a lattice with nearest-neighbor interactions. What are the speed limits on information propagation in quantum systems with power-law interactions, which decay as 1/r^{α} at distance r? Here, we present a definitive answer to this question for all exponents α>2d and all spatial dimensions d. Schematically, information takes time at least r^{min{1,α-2d}} to propagate a distance r. As recent state transfer protocols saturate this bound, our work closes a decades-long hunt for optimal Lieb-Robinson bounds on quantum information dynamics with power-law interactions.
The Lieb-Robinson correlation function can be used to characterize the propagation of quantum information in Ising chains. Considerable work has been done to establish bounds on this correlation function in various circumstances. To actually calculate the value of the correlation function directly typically requires a state space which grows exponentially with system size, and so is intractable for all but relatively small systems. We employ a recently-developed method that enables direct calculation of the Lieb--Robinson correlation function and which scales linearly with system size. This enables the computation for systems with more than a thousand qubits, revealing the propagation of quantum information down the chain. We extend this technique to the problem of Ising chains with randomly disordered coupling strengths. Increasing disorder causes localization of the quantum correlations and halts propagation of quantum information down the chain. We see light-cone isocontours therefore becoming vertical. The decay of correlations down the chain is characterized by a correlation length which in the paramagnetic phase increases quadratically with mean coupling strength and inverse quadratically with disorder level. The correlation length peaks at the phase transition. We map our general expression for the correlation function in the Ising system onto the probability density in the single particle tight-binding Su-Schrieffer-Heeger model for a specific initial condition.
Lieb-Robinson bounds quantify the maximal speed of information spreading in nonrelativistic quantum systems.We discuss the relation of Lieb-Robinson bounds to out-of-time order correlators, which correspond to different norms of commutators C(r, t ) = [A i (t ), B i+r ] of local operators.Using an exact Krylov space-time evolution technique, we calculate these two different norms of such commutators for the spin-1/2 Heisenberg chain with interactions decaying as a power law 1/r α with distance r.Our numerical analysis shows that both norms (operator norm and normalized Frobenius norm) exhibit the same asymptotic behavior, namely, a linear growth in time at short times and a power-law decay in space at long distance, leading asymptotically to power-law light cones for α < 1 and to linear light cones for α > 1.The asymptotic form of the tails of C(r, t ) ∝ t/r α is described by short-time perturbation theory, which is valid at short times and long distances.
We investigate how information spreads in three paradigmatic one-dimensional models with spatial disorder. The models we consider are unitarily related to a system of free fermions and, thus, are manifestly integrable. We demonstrate that out-of-time-order correlators can spread slowly beyond the single-particle localization length, despite the absence of many-body interactions. This phenomenon is shown to be due to the nonlocal relationship between elementary excitations and the physical degrees of freedom. We argue that this nonlocality becomes relevant for time-dependent correlation functions. In addition, a slow logarithmic-in-time growth of the entanglement entropy is observed following a quench from an unentangled initial state. We attribute this growth to the presence of strong zero modes, which gives rise to an exponential hierarchy of time scales upon ensemble averaging. Our work on disordered integrable systems complements the rich phenomenology of information spreading and we discuss broader implications for general systems with nonlocal correlations.
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.