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arXiv:2405.16842 · Physical review. B./Physical review. B

Locality, correlations, information, and non-Hermitian quantum systems

Brian Barch

Published Sep 6, 2024• 6 citations
arXivHTMLPDF

Abstract

Local non-Hermitian (NH) quantum systems generically exhibit breakdown of Lieb-Robinson (LR) bounds, motivating study of whether new locality measures might shed light not seen by existing measures. In this paper we discuss extensions of the connected correlation function (CC) as measures of locality and information spreading in both Hermitian and NH systems. We find that in Hermitian systems, $\ensuremath{\delta}\ensuremath{\rho}=\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}_{A}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{B}$ can be written as a linear combination of CCs, allowing placement of an LR bound on ${\ensuremath{\parallel}\ensuremath{\delta}\ensuremath{\rho}\ensuremath{\parallel}}_{2}$, which we show generically extends to an LR bound on mutual information. Additionally, we extend the CC to NH systems in a form that recovers locality, and use the metric formalism to derive a modified CC which recovers not just locality but even LR bounds in local $PT$-Symmetric systems. We find that even with these CCs, the bound on ${\ensuremath{\parallel}\ensuremath{\delta}\ensuremath{\rho}\ensuremath{\parallel}}_{2}$ breaks down in certain NH cases, which can be used to place a necessary condition on which local NH Hamiltonians are capable of nonlocal entanglement generation. Numerical simulations are provided by means of exact diagonalization for the NH Transverse-Field Ising Model, demonstrating both breakdown and recovery of LR bounds.

DOI
10.1103/physrevb.110.094307

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In weakly monitored quantum systems, traditional diagnostics of operator spreading such as out-of-time-ordered-correlators and Lieb-Robinson bounds can break down, as entanglement can spread at seemingly unbounded velocity. This has the adverse effect of making integrable and chaotic quantum dynamics appear identical. However, as the authors show here, the humble quantity of global operator entanglement remains a reliable metric of chaoticity in these systems. They elucidate this by studying entanglement phase transitions in a local, non-Hermitian transverse-field Ising model under the lens of operator and quenched entanglement dynamics.
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.
According to von Neumann, the global Hamiltonian of the whole universe must be Hermitian in order to keep the eigenvalues real and to construct a self-consistent quantum theory. In addition to the open system approach by introducing environmental degrees of freedom to a small system, a global Hermitian Hamiltonian can also be generated through the dilation from a small Hilbert space. For example, a local non-Hermitian PT\mathcal{P}TPT-symmetric system can be simulated with a global Hermitian one by the Naimark dilation. Recently, by introducing local measurements and investigating the correlation functions of outcomes, the internal nonlocality in such dilated Hermitian systems has been revealed, but only for a special case with a twofold structure. In this paper, we extend such a discussion to the generalized case when the twofold structure breaks. The internal nonlocality is discussed with different correlation pictures and the corresponding correlation bounds. Our results provide a device-independent test of the reliability of the simulation in the global Hermiticity.
The Lieb-Robinson bound states that local Hamiltonian evolution in nonrelativistic quantum mechanical theories gives rise to the notion of an effective light cone with exponentially decaying tails. We discuss several consequences of this result in the context of quantum information theory. First, we show that the information that leaks out to spacelike separated regions is negligible and that there is a finite speed at which correlations and entanglement can be distributed. Second, we discuss how these ideas can be used to prove lower bounds on the time it takes to convert states without topological quantum order to states with that property. Finally, we show that the rate at which entropy can be created in a block of spins scales like the boundary of that block.
It has been suggested that there may exist quantum correlations that go beyond entanglement. The existence of such correlations can be revealed by information theoretic quantities such as quantum discord, but not by the conventional measures of entanglement. We argue that a state displays quantumness, that can be of local and nonlocal origin. Information theoretic measures not only characterize the nonlocal quantumness, but also the local quantumness, such as the “local superposition”. This can be a reason, why such measures are nonzero, when there is no entanglement. We consider a generalized version of the Werner state to demonstrate the interplay of local quantumness, nonlocal quantumness and classical mixedness of a state.
The Lieb-Robinson (LR) bound rigorously shows that in quantum systems with short-range interactions, the maximum amount of information that travels beyond an effective "light cone" decays exponentially with distance from the light-cone front, which expands at finite velocity.Despite being a fundamental result, existing bounds are often extremely loose, limiting their applications.We introduce a method that dramatically and qualitatively improves LR bounds in models with finite-range interactions.Most prominently, in systems with a large local Hilbert space dimension D, our method gives a LR velocity that grows much slower than previous bounds with D as D → ∞.For example, in the Heisenberg model with spin S, we find v ≤ const.compared to the previous v ∝ S, which diverges at large S, and in multiorbital Hubbard models with N orbitals, we find v ∝ √ N instead of previous v ∝ N , and similarly in the Nstate truncated Bose-Hubbard model and Wen's quantum rotor model.Our bounds also scale qualitatively better in some systems when the spatial dimension or certain model parameters become large, for example in the d-dimensional quantum Ising model and perturbed toric code models.Even in spin-1/2 Ising and Fermi-Hubbard models, our method improves the LR velocity by an order of magnitude with typical model parameters, and significantly improves the LR bound at large distance and early time.