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arXiv:1806.00113 · Physical review. E

Tripartite mutual information, entanglement, and scrambling in permutation symmetric systems with an application to quantum chaos

Akshay Seshadri, Vaibhav Madhok, Arul Lakshminarayan

Published Nov 6, 2018• 64 citations
arXivHTMLPDF

Abstract

Many-body states that are invariant under particle relabeling, the permutation symmetric states, occur naturally when the system dynamics is described by symmetric processes or collective spin operators. We derive expressions for the reduced density matrix for arbitrary subsystem decomposition for these states and study properties of permutation symmetric states and their subsystems when the joint system is picked randomly and uniformly. Thus defining an appropriate random matrix ensemble, we find the average linear entropy and von Neumann entropy, which implies that random permutation symmetric states are marginally entangled and as a consequence the tripartite mutual information (TMI) is typically positive, preventing information from being shared globally. Applying these results to the quantum kicked top viewed as a multiqubit system, we find that entanglement, mutual information, and TMI all increase for large subsystems across the Ehrenfest or logarithmic time and saturate at the random state values if there is global chaos. During this time the out-of-time-order correlators evolve exponentially, implying scrambling in phase space. We discuss how positive TMI may coexist with such scrambling.

DOI
10.1103/physreve.98.052205

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Many-body states that are invariant under particle relabeling, the permutation symmetric states, occur naturally when the system dynamics is described by symmetric processes or collective spin operators. We derive expressions for the reduced density matrix for arbitrary subsystem decomposition for these states and study properties of permutation symmetric states and their subsystems when the joint system is picked randomly and uniformly. Thus defining an appropriate random matrix ensemble, we find the average linear entropy and von Neumann entropy, which implies that random permutation symmetric states are marginally entangled and as a consequence the tripartite mutual information (TMI) is typically positive, preventing information from being shared globally. Applying these results to the quantum kicked top viewed as a multiqubit system, we find that entanglement, mutual information, and TMI all increase for large subsystems across the Ehrenfest or logarithmic time and saturate at the random state values if there is global chaos. During this time the out-of-time-order correlators evolve exponentially, implying scrambling in phase space. We discuss how positive TMI may coexist with such scrambling.
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.
Tripartite mutual information (TMI) is an efficient observable to quantify the ability of a scrambler for the unitary time-evolution operator with a quenched many-body Hamiltonian. In this paper we give numerical demonstrations of the TMI in disorder-free (translationally invariant) spin models with three-body and four-body multiple-spin interactions. The dynamical behavior of the TMI of these models does not exhibit a linear light cone for sufficiently strong interactions. In early-time evolution, the TMI displays a distinct negative increase fitted by a logarithmiclike function. This is in contrast to the conventional linear light-cone behavior present in the XXZXXZXXZ model and its near-integrable vicinity. The late-time evolution of the TMI in finite-size systems is also numerically investigated. The multiple-spin interactions make the system nearly integrable and weakly suppress the spread of information and scrambling. The observation of the late-time value of the TMI indicates that the scrambling nature of the system changes by interactions and this change can be characterized by a phase-transition-like behavior of the TMI, reflecting the system's integrability and violating the eigenstate thermalization hypothesis.
Studying the typical entanglement entropy of a bipartite system when averaging over different ensembles of pure quantum states has been instrumental in different areas of physics, ranging from many-body quantum chaos to black hole evaporation. We extend such analysis to open quantum systems and mixed states, where we compute the typical mutual information in a bipartite system averaged over the ensemble of mixed Gaussian states with a fixed spectrum. Tools from random matrix theory and determinantal point processes allow us to compute arbitrary k-point correlation functions of the singular values of the corresponding complex structure in a subsystem for a given spectrum in the full system. In particular, we evaluate the average von Neumann entropy in a subsystem based on the level density and the average mutual information. Those results are given for finite system size as well as in the thermodynamic limit.
The projected ensemble -- an ensemble of pure states on a subsystem conditioned on projective measurement outcomes on its complement -- provides a finer probe of ergodicity and information structure than the reduced density matrix of the subsystem in bipartite quantum states. This framework can be generalised to partial projected ensembles in tripartite settings, where outcomes from part of the measured subsystem are discarded, leading to ensembles of mixed states. We show that information measures defined for such ensembles, in particular the Holevo information, yield a more detailed characterisation of how quantum information is distributed between subsystems compared to conventional entanglement measures. Using exact analytical results supported by numerical results, we uncover a qualitative change in the scaling of the Holevo information with system size in partial projected ensembles generated by Haar-random states, as the relative sizes of the subsystem are varied. In one phase, the Holevo information decays exponentially with system size, while in the other it grows linearly, thereby defining distinct information phases separated by sharp transitions signalled by non-analyticities in the Holevo information. The exponentially decaying phase rigorously establishes the existence of a measurement-invisible quantum-correlated phase -- a manifestation of many-body information scrambling with no bipartite analogue. We contrast this information-phase diagram with the entanglement-phase structure of tripartite Haar-random states obtained from logarithmic negativity, and show that the Holevo information reveals additional fine structure beyond conventional entanglement measures. Finally, we show that these information phases, as characterised by the Holevo information, emerge in the dynamics of chaotic quantum circuits and discuss the associated timescales.
We study chaos and scrambling in unitary channels by considering their entanglement properties as states. Using out-of-time-order correlation functions to diagnose chaos, we characterize the ability of a channel to process quantum information. We show that the generic decay of such correlators implies that any input subsystem must have near vanishing mutual information with almost all partitions of the output. Additionally, we propose the negativity of the tripartite information of the channel as a general diagnostic of scrambling. This measures the delocalization of information and is closely related to the decay of out-of-time-order correlators. We back up our results with numerics in two non-integrable models and analytic results in a perfect tensor network model of chaotic time evolution. These results show that the butterfly effect in quantum systems implies the information-theoretic definition of scrambling.
Information scrambling is a hallmark of quantum chaos and thermalization in isolated quantum many-body systems. We investigate scrambling dynamics in the all-to-all interacting spin Sachdev-Ye-Kitaev (SYK)-qqq model using both pure- and mixed-state entanglement measures. We show that von-Neumann and Rényi entropies exhibit rapid growth followed by saturation near Haar-random values, signaling efficient scrambling. The scrambling rate reveals a nontrivial dependence on the interaction order, system size, and Hamiltonian scaling. We further employ mixed-state entanglement as a powerful probe of information scrambling. We numerically find a universal relation between the Rényi-1/2 mutual information and entanglement negativity for minimal interaction order in the early growth regime. Furthermore, entanglement negativity displays a Page-curve-like behavior under unequal subsystem partitioning, characterized by the birth, spread, and eventual death of quantum correlations. Our results provide a generic description of information scrambling using entanglement dynamics in all-to-all interacting spin systems with multi-body interactions.