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arXiv:1907.07637 · Physical Review Letters

Finite Speed of Quantum Scrambling with Long Range Interactions

Chi-Fang Chen, Andrew Lucas

Published Dec 20, 2019• 118 citations
arXivHTMLPDF

Abstract

In a locally interacting many-body system, two isolated qubits, separated by a large distance r, become correlated and entangled with each other at a time t≥r/v. This finite speed v of quantum information scrambling limits quantum information processing, thermalization, and even equilibrium correlations. Yet most experimental systems contain long range power-law interactions-qubits separated by r have potential energy V(r)∝r^{-α}. Examples include the long range Coulomb interactions in plasma (α=1) and dipolar interactions between spins (α=3). In one spatial dimension, we prove that the speed of quantum scrambling remains finite for sufficiently large α. This result parametrically improves previous bounds, compares favorably with recent numerical simulations, and can be realized in quantum simulators with dipolar interactions. Our new mathematical methods lead to improved algorithms for classically simulating quantum systems, and improve bounds on environmental decoherence in experimental quantum information processors.

DOI
10.1103/physrevlett.123.250605

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In a locally interacting many-body system, two isolated qubits, separated by a large distance r, become correlated and entangled with each other at a time t≥r/v. This finite speed v of quantum information scrambling limits quantum information processing, thermalization, and even equilibrium correlations. Yet most experimental systems contain long range power-law interactions-qubits separated by r have potential energy V(r)∝r^{-α}. Examples include the long range Coulomb interactions in plasma (α=1) and dipolar interactions between spins (α=3). In one spatial dimension, we prove that the speed of quantum scrambling remains finite for sufficiently large α. This result parametrically improves previous bounds, compares favorably with recent numerical simulations, and can be realized in quantum simulators with dipolar interactions. Our new mathematical methods lead to improved algorithms for classically simulating quantum systems, and improve bounds on environmental decoherence in experimental quantum information processors.
In ergodic many-body quantum systems, locally encoded quantum information becomes, in the course of time evolution, inaccessible to local measurements. This concept of "scrambling" is currently of intense research interest, entailing a deep understanding of many-body dynamics such as the processes of chaos and thermalization. Here, we present first experimental demonstrations of quantum information scrambling on a 10-qubit trapped-ion quantum simulator representing a tunable long-range interacting spin system, by estimating out-of-time ordered correlators (OTOCs) through randomized measurements. We also analyze the role of decoherence in our system by comparing our measurements to numerical simulations and by measuring Rényi entanglement entropies.
Strongly long-range interacting quantum systems---those with interactions decaying as a power law $1/{r}^{\ensuremath{\alpha}}$ in the distance rrr on a DDD-dimensional lattice for $\ensuremath{\alpha}\ensuremath{\le}D$---have received significant interest in recent years. They are present in leading experimental platforms for quantum computation and simulation, as well as in theoretical models of quantum-information scrambling and fast entanglement creation. Since no notion of locality is expected in such systems, a general understanding of their dynamics is lacking. In a step towards rectifying this problem, we prove two Lieb-Robinson-type bounds that constrain the time for signaling and scrambling in strongly long-range interacting systems, for which no tight bounds were previously known. Our first bound applies to systems mappable to free-particle Hamiltonians with long-range hopping, and is saturable for $\ensuremath{\alpha}\ensuremath{\le}D/2$. Our second bound pertains to generic long-range interacting spin Hamiltonians and gives a tight lower bound for the signaling time to extensive subsets of the system for all $\ensuremath{\alpha}<D$. This many-site signaling time lower bounds the scrambling time in strongly long-range interacting systems.
In this study, we investigate out-of-time-order correlators (OTOCs) in systems with power-law decaying interactions such as ${R}^{\ensuremath{-}\ensuremath{\alpha}}$, where RRR is the distance. In such systems, the fast scrambling of quantum information or the exponential growth of information propagation can potentially occur according to the decay rate $\ensuremath{\alpha}$. In this regard, a crucial open challenge is to identify the optimal condition for $\ensuremath{\alpha}$ such that fast scrambling cannot occur. In this study, we disprove fast scrambling in generic long-range interacting systems with $\ensuremath{\alpha}>D$ (DDD: spatial dimension), where the total energy is extensive in terms of system size and the thermodynamic limit is well defined. We rigorously demonstrate that the OTOC shows a polynomial growth over time as long as $\ensuremath{\alpha}>D$ and the necessary scrambling time over a distance RRR is larger than $t\ensuremath{\gtrsim}{R}^{[(2\ensuremath{\alpha}\ensuremath{-}2D)/(2\ensuremath{\alpha}\ensuremath{-}D+1)]}$.
Scrambling, the delocalization of initially localized quantum information, is commonly characterized by the out-of-time ordered correlator (OTOC). Employing the OTOC-Renyi-2 entropy theorem we derive a quantum speed limit for the OTOC, which sets an lower bound for the rate with which information can be scrambled. This bound becomes particularly tractable by describing the scrambling of information in a closed quantum system as an effective decoherence process of an open system interacting with an environment. We prove that decay of the OTOC can be bounded by the strength of the system-environment coupling and two-point environmental correlation functions. We validate our analytic bound numerically using the non-integrable transverse field Ising model. Our results provide a universal and model-agnostic quantitative framework for understanding the dynamical limits of information spreading across quantum many-body physics, condensed matter, and engineered quantum platforms.
We study quantum information scrambling in spin models with both long-range all-to-all and short-range interactions. We argue that a simple global, spatially homogeneous interaction together with local chaotic dynamics is sufficient to give rise to fast scrambling, which describes the spread of quantum information over the entire system in a time that is logarithmic in the system size. This is illustrated in two tractable models: (1) a random circuit with Haar random local unitaries and a global interaction and (2) a classical model of globally coupled nonlinear oscillators. We use exact numerics to provide further evidence by studying the time evolution of an out-of-time-order correlator and entanglement entropy in spin chains of intermediate sizes. Our results pave the way towards experimental investigations of fast scrambling and aspects of quantum gravity with quantum simulators.
The nonequilibrium response of a quantum many-body system defines its fundamental transport properties and how initially localized quantum information spreads. However, for long-range-interacting quantum systems little is known. We address this issue by analyzing a local quantum quench in the long-range Ising model in a transverse field, where interactions decay as a variable power law with distance ∝r(-α), α>0. Using complementary numerical and analytical techniques, we identify three dynamical regimes: short-range-like with an emerging light cone for α>2, weakly long range for 1<α<2 without a clear light cone but with a finite propagation speed of almost all excitations, and fully nonlocal for α<1 with instantaneous transmission of correlations. This last regime breaks generalized Lieb-Robinson bounds and thus locality. Numerical calculation of the entanglement spectrum demonstrates that the usual picture of propagating quasiparticles remains valid, allowing an intuitive interpretation of our findings via divergences of quasiparticle velocities. Our results may be tested in state-of-the-art trapped-ion experiments.