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arXiv:0805.4149 · Physical Review A

Complexity of thermal states in quantum spin chains

Marko Žnidarič, Tomaž Prosen, Iztok Pižorn

Published Aug 11, 2008• 68 citations
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Abstract

We study the quantum correlations and complexity of simulation, characterized by quantum mutual information and entanglement entropy in operator space, respectively, for thermal states in critical, noncritical, and quantum chaotic spin chains. A simple general relation between the two quantities is proposed. We show that in all cases mutual information and entanglement entropy saturate with system size, whereas as a function of the inverse temperature, we find logarithmic divergences for critical cases and uniform bounds in noncritical cases. A simple efficient quasiexact method for computation of arbitrary entropy-related quantities in thermalized $XY$ spin chains is proposed.

DOI
10.1103/physreva.78.022103

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We study the quantum correlations and complexity of simulation, characterized by quantum mutual information and entanglement entropy in operator space, respectively, for thermal states in critical, noncritical, and quantum chaotic spin chains. A simple general relation between the two quantities is proposed. We show that in all cases mutual information and entanglement entropy saturate with system size, whereas as a function of the inverse temperature, we find logarithmic divergences for critical cases and uniform bounds in noncritical cases. A simple efficient quasiexact method for computation of arbitrary entropy-related quantities in thermalized XYXYXY spin chains is proposed.
Entanglement properties of excited eigenstates (or of thermal mixed states) are difficult to study with conventional analytical methods. We approach this problem for random spin chains using a recently developed real-space renormalization group technique for excited states (``RSRG-X''). For the random XX and quantum Ising chains, which have logarithmic divergences in the entanglement entropy of their (infinite-randomness) critical ground states, we show that the entanglement entropy of excited eigenstates retains a logarithmic divergence while the mutual information of thermal mixed states does not. However, in the XX case the coefficient of the logarithmic divergence extends from the universal ground-state value to a universal interval due to the degeneracy of excited eigenstates. These models are noninteracting in the sense of having free-fermion representations, allowing strong numerical checks of our analytical predictions.
Characterizing complexity and criticality in quantum systems requires diagnostics that are both computationally tractable and physically insightful. We apply a measure of quantum state complexity for n-qubit systems, defined as the divergence between the Shannon or von Neumann entropy of the computational basis distribution and the second-order Renyi entropy. This quantity has already been used earlier termed as structural entropy and it is particularly powerful as the Renyi entropy is directly related to state purity, linear entropy, and the inverse participation ratio, providing a clear physical grounding. While other Renyi orders could be used, the second order offers a deep and established connection to these key physical quantities. We first validate the measure in canonical noise channels, showing it peaks at the boundary between quantum and classical regimes. We then demonstrate its power in many-body physics. For systems exhibiting a many-body localization transition - including deformed random matrix ensembles and a disordered Heisenberg spin chain - the complexity measure reliably signals the crossover from integrable/localized to quantum-chaotic/ergodic phases. Crucially, the maximum complexity occurs in the non-ergodic yet extended states at the transition, precisely capturing the critical region where the system is neither fully localized nor thermalized. Furthermore, within the chaotic phase, the measure correlates with the survival probability of local excitations, revealing a spectrum of thermalization properties. Our results establish that the entropic complexity is a simple, versatile, and effective probe for identifying nontrivial quantum regimes and transitions giving a new and alternative insight into such systems.
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The complexity of representation of operators in quantum mechanics can be characterized by the operator space entanglement entropy (OSEE). We show that in the homogeneous Heisenberg XYXYXY spin 1/2 chains the OSEE for initial local operators grows at most logarithmically with time. The prefactor in front of the logarithm generally depends only on the number of stationary points of the quasiparticle dispersion relation and for the XYXYXY model changes from 1/3 to 2/3 exactly at the point of quantum phase transition to long-range magnetic correlations in the nonequilibrium steady state. In addition, we show that the presence of a small disorder triggers a saturation of the OSEE.
We consider the single-copy entanglement as a quantity to assess quantum correlations in the ground state in quantum many-body systems. We show for a large class of models that already on the level of single specimens of spin chains, criticality is accompanied with the possibility of distilling a maximally entangled state of arbitrary dimension from a sufficiently large block deterministically, with local operations and classical communication. These analytical results---which refine previous results on the divergence of block entropy as the rate at which maximally entangled pairs can be distilled from many identically prepared chains---are made quantitative for general isotropic translationally invariant spin chains that can be mapped onto a quasifree fermionic system, and for the anisotropic XYXYXY model. For the XXXXXX model, we provide the asymptotic scaling of $\ensuremath{\sim}(1∕6){\mathrm{log}}_{2}(L)$, and contrast it with the block entropy.
We study the generalized mutual information ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{I}}_{n}$ of the ground state of different critical quantum chains. The generalized mutual information definition that we use is based on the well established concept of the R\'enyi divergence. We calculate this quantity numerically for several distinct quantum chains having either discrete Z(Q)Z(Q)Z(Q) symmetries (QQQ-state Potts model with Q=2,3,4Q=2,3,4Q=2,3,4 and Z(Q)Z(Q)Z(Q) parafermionic models with Q=5,6,7,8Q=5,6,7,8Q=5,6,7,8 and also Ashkin-Teller model with different anisotropies) or the U(1)U(1)U(1) continuous symmetries (Klein-Gordon field theory, XXZXXZXXZ and spin-1 Fateev-Zamolodchikov quantum chains with different anisotropies). For the spin chains these calculations were done by expressing the ground-state wave functions in two special bases. Our results indicate some general behavior for particular ranges of values of the parameter nnn that defines ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{I}}_{n}$. For a system, with total size LLL and subsystem sizes $\ensuremath{\ell}$ and $L\ensuremath{-}\ensuremath{\ell}$, the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{I}}_{n}$ has a logarithmic leading behavior given by $\frac{{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}}_{n}}{4}log[\frac{L}{\ensuremath{\pi}}sin(\frac{\ensuremath{\pi}\ensuremath{\ell}}{L})]$ where the coefficient ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}}_{n}$ is linearly dependent on the central charge ccc of the underlying conformal field theory describing the system's critical properties.