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arXiv:1808.05646 · Physical review. B./Physical review. B

Tripartite information, scrambling, and the role of Hilbert space partitioning in quantum lattice models

Oskar H Schnaack, Niklas Bölter, Sebastian Paeckel, Salvatore R. Manmana, Stefan K. Kehrein, Markus Schmitt

Published Dec 11, 2019• 41 citations
arXivHTMLPDF

Abstract

For the characterization of the dynamics in quantum many-body systems the question of how information spreads and becomes distributed over the constituent degrees of freedom is of fundamental interest. The delocalization of information under many-body dynamics has been dubbed scrambling, and out-of-time-order correlators were proposed to probe this behavior. In this work we investigate the time evolution of tripartite information as a natural operator-independent measure of scrambling, which quantifies to what extent the initially localized information can be recovered only by global measurements. Studying the dynamics of quantum lattice models with tunable integrability breaking, we demonstrate that in contrast to quadratic models generic interacting systems scramble information irrespective of the chosen partitioning of the Hilbert space, which justifies the characterization as a scrambler. Without interactions the dynamics of tripartite information in momentum space reveals unambiguously the absence of scrambling.

DOI
10.1103/physrevb.100.224302

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For the characterization of the dynamics in quantum many-body systems the question of how information spreads and becomes distributed over the constituent degrees of freedom is of fundamental interest. The delocalization of information under many-body dynamics has been dubbed scrambling, and out-of-time-order correlators were proposed to probe this behavior. In this work we investigate the time evolution of tripartite information as a natural operator-independent measure of scrambling, which quantifies to what extent the initially localized information can be recovered only by global measurements. Studying the dynamics of quantum lattice models with tunable integrability breaking, we demonstrate that in contrast to quadratic models generic interacting systems scramble information irrespective of the chosen partitioning of the Hilbert space, which justifies the characterization as a scrambler. Without interactions the dynamics of tripartite information in momentum space reveals unambiguously the absence of scrambling.
This thesis contributes to the study of quantum chaos and thermalization of closed quantum systems by applying tools from quantum information theory. We study scrambling – the delocalization of quantum information – in one-dimensional quantum many-body systems using the observable-independent tripartite information 𝐼3. We consider closed quantum systems that undergo unitary time evolution governed by a Hamiltonian, and compare them to the strong scrambling caused by random unitary operators (Haar scrambling). The tripartite information is a tool from quantum information theory that allows us to quantify correlations in space and time by considering the unitary operator evolving the system as a state |𝑈 (𝑡)⟩ in a doubled Hilbert space H ⊗ H. We found a ballistic spreading of information in translation invariant systems of spinless fermions that causes Haar scrambling in interacting models. For strongly disordered versions of the XXZ chain and the transverse-field Ising chain we found a complete breakdown of in- formation transport in non-interacting Anderson localized systems, as well as a logarithmically slow spreading of the information signal in systems that are said to be many-body localized (MBL).
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
We systematically investigate scrambling (or delocalizing) processes of quantum information encoded in quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling, we adopt the tripartite mutual information (TMI) that becomes negative when quantum information is delocalized. We clarify that scrambling is an independent property of the integrability of Hamiltonians; TMI can be negative or positive for both integrable and nonintegrable systems. This implies that scrambling is a separate concept from conventional quantum chaos characterized by nonintegrability. Specifically, we argue that there are a few exceptional initial states that do not exhibit scrambling, and show that such exceptional initial states have small effective dimensions. Furthermore, we calculate TMI in the Sachdev-Ye-Kitaev (SYK) model, a fermionic toy model of quantum gravity. We find that disorder does not make scrambling slower but makes it smoother in the SYK model, in contrast to many-body localization in spin chains.
The delocalization or scrambling of quantum information has emerged as a central ingredient in the understanding of thermalization in isolated quantum many-body systems. Recently, significant progress has been made analytically by modeling non-integrable systems as periodically driven systems, lacking a Hamiltonian picture, while honest Hamiltonian dynamics are frequently limited to small system sizes due to computational constraints. In this paper, we address this by investigating the role of conservation laws (including energy conservation) in the thermalization process from an information-theoretic perspective. For general non-integrable models, we use the equilibrium approximation to show that the maximal amount of information is scrambled (as measured by the tripartite mutual information of the time-evolution operator) at late times even when a system conserves energy. In contrast, we explicate how when a system has additional symmetries that lead to degeneracies in the spectrum, the amount of information scrambled must decrease. This general theory is exemplified in case studies of holographic conformal field theories (CFTs) and the Sachdev-Ye-Kitaev (SYK) model. Due to the large Virasoro symmetry in 1+1D CFTs, we argue that, in a sense, these holographic theories are not maximally chaotic, which is explicitly seen by the non-saturation of the second Rényi tripartite mutual information. The roles of particle-hole and U(1) symmetries in the SYK model are milder due to the degeneracies being only two-fold, which we confirm explicitly at both large- and small-N. We reinterpret the operator entanglement in terms of the growth of local operators, connecting our results with the information scrambling described by out-of-time-ordered correlators, identifying the mechanism for suppressed scrambling from the Heisenberg perspective.
Quantum information scrambling has attracted much attention amid the effort to reconcile the conflict between quantum-mechanical unitarity and the thermalization irreversibility in many-body systems. Here we propose an unconventional mechanism to generate quantum information scrambling through a high-complexity mapping from logical to physical degrees-of-freedom that hides the logical information into nonseparable many-body correlations. Corresponding to this mapping, we develop an algorithm to efficiently sample a Slater-determinant wave function and compute all physical observables in dynamics with a polynomial cost in system size. The system shows information scrambling in the quantum many-body Hilbert space characterized by the spreading of Hamming distance. At late time we find emergence of classical diffusion dynamics in this quantum many-body system. We establish that the operator mapping enabled growth in an out-of-time-order correlator exhibits exponential-scrambling behavior. The quantum information-hiding mapping approach may shed light on the understanding of fundamental connections among computational complexity, information scrambling, and quantum thermalization.
A bstract We consider the Brownian SYK model of N interacting Majorana fermions, with random couplings that are taken to vary independently at each time. We study the out-of-time-ordered correlators (OTOCs) of arbitrary observables and the Rényi-2 tripartite information of the unitary evolution operator, which were proposed as diagnostic tools for quantum chaos and scrambling, respectively. We show that their averaged dynamics can be studied as a quench problem at imaginary times in a model of N qudits, where the Hamiltonian displays site-permutational symmetry. By exploiting a description in terms of bosonic collective modes, we show that for the quantities of interest the dynamics takes place in a subspace of the effective Hilbert space whose dimension grows either linearly or quadratically with N , allowing us to perform numerically exact calculations up to N = 10 6 . We analyze in detail the interesting features of the OTOCs, including their dependence on the chosen observables, and of the tripartite information. We observe explicitly the emergence of a scrambling time t ∗ ∼ ln N controlling the onset of both chaotic and scrambling behavior, after which we characterize the exponential decay of the quantities of interest to the corresponding Haar scrambled values.