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Quantum Dynamical Entropy of Spin Systems

Takayuki Miyadera, Masanori Ohya

Published Jan 1, 2003• 2 citations
OpenAlex

Abstract

We investigate a quantum dynamical entropy of one-dimesional quantum spin systems. We show that the dynamical entropy is bounded from above by a quantity which is related with group velocity determined by the interaction and mean entropy of the state.

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This paper is a contribution to the general theory of quantum spin systems. We deal with a general theory in the sense that no concrete interaction shows up but an arbitrary relatively short-range interaction is chosen. It is well known that the mean entropy plays an important role in the thermodynamics of quantum spin systems: it is one of the ingredients of the Lanford–Ruelle–Robinson variational principle. We show that in the background of the existence of the mean entropy, there is an operator convergence which resembles the McMillan theorem from information theory. Asymptotic equipartition property and several entropy densities are investigated, in particular, the relation of the Gibbs condition to relative entropies. Although this paper is not intended to be a review, we try to give an overview of the theory of quantum Gibbs states in the mathematical sense. The paper is organized as follows. Section 1 provides an introduction and further sections contain our new results. Each section is closed with a short discussion on sources and references.
We review some of the recent progress on the study of entropy of entanglement in many-body quantum systems. Emphasis is placed on the scaling properties of entropy for one-dimensional multi-partite models at quantum phase transitions and, more generally, on the concept of area law. We also briefly describe the relation between entanglement and the presence of impurities, the idea of particle entanglement, the evolution of entanglement along renormalization group trajectories, the dynamical evolution of entanglement and the fate of entanglement along a quantum computation.
We study the entanglement property in matrix product spin-ring systems systemically by von Neumann entropy. We find that: (i) the Hilbert space dimension of one spin determines the upper limit of the maximal value of the entanglement entropy of one spin, while for multiparticle entanglement entropy, the upper limit of the maximal value depends on the dimension of the representation matrices. Based on the theory, we can realize the maximum of the entanglement entropy of any spin block by choosing the appropriate control parameter values. (ii) When the entanglement entropy of one spin takes its maximal value, the entanglement entropy of an asymptotically large spin block, i.e. the renormalization group fixed point, is not likely to take its maximal value, and so only the entanglement entropy S n of a spin block that varies with size n can fully characterize the spin-ring entanglement feature. Finally, we give the entanglement dynamics, i.e. the Hamiltonian of the matrix product system.
Traditional thermodynamic trade-off relations usually apply to quantities that depend linearly on probability distributions. In contrast, many important information-theoretic measures, such as entropies, are nonlinear and therefore difficult to analyze with existing frameworks. Motivated by replica methods in quantum information and spin-glass theory, we introduce replica Markov processes, i.e., Markovian dynamics of KKK independent, identical copies, and derive trade-off relations that bound relative moments of replica observables in terms of the dynamical activity. By choosing appropriate replica observables, these inequalities translate into bounds on nonlinear quantities of the original single process. Focusing on entropic measures of uncertainty, we obtain upper bounds on both the time derivative and the value of the Tsallis entropy for general trajectory-observable distributions. Moreover, we derive upper bounds on the Rényi and Tsallis entropies, where the bounds involve the initial dynamical activity. Analogous bounds can be derived for the entropies of the state distribution. In particular, we provide upper bounds on both the time derivative and the value of the Tsallis entropy of state distributions. Moreover, we show that the Rényi and Tsallis entropies of the state distribution are bounded from above by terms involving the local escape rate from the initial node. We also illustrate how replicas can be used to study other nonlinear functionals such as extreme-value observables. Finally, we extend the construction to continuously monitored open quantum systems, where the bounds are expressed in terms of the quantum dynamical activity. These results provide entropic counterparts to activity-based uncertainty relations and establish a general method for constraining nonlinear information-theoretic quantities in stochastic and quantum thermodynamics.
Entropy, and its temporal evolution, play a central role in the foundations of quantum theory and in modern quantum technologies. Here we study, in particular, the relations between the --- in general, non-Markovian --- evolution of an open quantum system, the notions of divisibility of a dynamical map and of distinguishability of quantum states, and the temporal behaviour of various entropy-related quantities such as the Renyi (and sandwiched Renyi) divergences, and the so-called min- and max- conditional entropies. This, in turn, gives rise to an operational meaning of (non-)Markovianity.