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arXiv:1612.08781 · Physical review. E

Out-of-time-order fluctuation-dissipation theorem

Naoto Tsuji, Tomohiro Shitara, Masahito Ueda

Published Jan 3, 2018• 32 citations
arXivHTMLPDF

Abstract

We prove a generalized fluctuation-dissipation theorem for a certain class of out-of-time-ordered correlators (OTOCs) with a modified statistical average, which we call bipartite OTOCs, for general quantum systems in thermal equilibrium. The difference between the bipartite and physical OTOCs defined by the usual statistical average is quantified by a measure of quantum fluctuations known as the Wigner-Yanase skew information. Within this difference, the theorem describes a universal relation between chaotic behavior in quantum systems and a nonlinear-response function that involves a time-reversed process. We show that the theorem can be generalized to higher-order n-partite OTOCs as well as in the form of generalized covariance.

DOI
10.1103/physreve.97.012101

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We prove a generalized fluctuation-dissipation theorem for a certain class of out-of-time-ordered correlators (OTOCs) with a modified statistical average, which we call bipartite OTOCs, for general quantum systems in thermal equilibrium. The difference between the bipartite and physical OTOCs defined by the usual statistical average is quantified by a measure of quantum fluctuations known as the Wigner-Yanase skew information. Within this difference, the theorem describes a universal relation between chaotic behavior in quantum systems and a nonlinear-response function that involves a time-reversed process. We show that the theorem can be generalized to higher-order n-partite OTOCs as well as in the form of generalized covariance.
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The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolutions, as given by quantum maps. Third, it brings out the full connection between the FDT and the Quantum Fisher information, the figure of merit in quantum metrology.
There is a remarkable interest in the study of out-of-time ordered correlators (OTOCs) that goes from many-body theory and high-energy physics to quantum chaos. In the latter case there is a special focus on the comparison with the traditional measures of quantum complexity such as the spectral statistics. The exponential growth has been verified for many paradigmatic maps and systems. However, less is known for multipartite cases. On the other hand, the recently introduced Wigner separability entropy (WSE) and its classical counterpart provide a complexity measure that treats equally quantum and classical distributions in phase space. We compare the behavior of these measures in a system consisting of two coupled and perturbed cat maps with different dynamics: Double hyperbolic, double elliptic, and mixed. In all cases, we find that the OTOCs and the WSE have essentially the same behavior, providing a complete characterization in generic bipartite systems and at the same time revealing them as very good measures of quantum complexity for phase-space distributions. Moreover, we establish a relation between both quantities by means of a recently proven theorem linking the second Rényi entropy and OTOCs.
We establish a fluctuation theorem for an open quantum bipartite system that explicitly manifests the role played by quantum correlation. Generally quantum correlations may substantially modify the universality of classical thermodynamic relations in composite systems. Our fluctuation theorem finds a non-equilibrium parameter of genuinely quantum nature that sheds light on the emerging quantum information thermodynamics. Specifically we show that the statistics of quantum correlation fluctuation obtained in a time-reversed process can provide a useful insight into addressing work and heat in the resulting thermodynamic evolution. We illustrate these quantum thermodynamic relations by two examples of quantum correlated systems.
The fluctuation-dissipation theorem (FDT) is a cornerstone of statistical mechanics, linking the linear response of a system to external perturbations with the equilibrium fluctuations of its constituent degrees of freedom. Traditionally formulated for Hermitian quantum systems in thermal equilibrium, its applicability to non-Hermitian quantum systems, which are increasingly relevant in areas such as open quantum systems, PT-symmetric quantum mechanics, and topological phases, remains a complex and largely unresolved challenge. This paper presents a theoretical framework for a universal fluctuation-dissipation theorem applicable to a broad class of non-Hermitian quantum systems. We demonstrate how the presence of non-Hermiticity, characterized by complex energy eigenvalues and non-orthogonal eigenstates, fundamentally modifies the relationship between correlation functions and response functions. Our derivation utilizes a generalized time-evolution operator and incorporates concepts from non-Hermitian spectral theory, revealing a modified FDT that accounts for energy gain or loss inherent in these systems. The resulting theorem provides insights into the interplay between quantum noise and dissipation in environments far from conventional thermal equilibrium, offering a crucial tool for understanding and engineering non-Hermitian quantum devices. We discuss the implications for non-equilibrium steady states and the breakdown of standard notions of temperature and entropy, proposing a path towards experimental verification in platforms like photonic lattices and cold atoms with controlled losses.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolution, as given by quantum maps. Third, it clarifies the connection between the FDT and quantum metrology in systems with a non-equilibrium steady state.