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arXiv:2104.05722 · International Journal of Quantum Information

Entropy of temporal entanglement

Leonardo Castellani

Published Nov 18, 2022• 2 citations
arXivHTMLPDF

Abstract

A recently proposed history formalism is used to define temporal entanglement in quantum systems, and compute its entropy. The procedure is based on the time-reduction of the history density operator, and allows a symmetrical treatment of space and time correlations. Temporal entanglement entropy is explicitly calculated in two simple quantum computation circuits.

DOI
10.1142/s0219749923500016

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A recently proposed history formalism is used to define temporal entanglement in quantum systems, and compute its entropy. The procedure is based on the time-reduction of the history density operator, and allows a symmetrical treatment of space and time correlations. Temporal entanglement entropy is explicitly calculated in two simple quantum computation circuits.
In this paper we present a concept of quantum entanglement in time in a context of entangled consistent histories. These considerations are supported by presentation of necessary tools closely related to those acting on a space of spatial multipartite quantum states. We show that in similarity to monogamy of quantum entanglement in space, quantum entanglement in time is also endowed with this property for a particular history. Basing on these observations, we discuss further bounding of temporal correlations and derive analytically the Tsirelson bound implied by entangled histories for the Leggett-Garg inequalities.
The concept of space evolution (or space-time duality) has emerged as a promising approach for studying quantum dynamics.The basic idea involves exchanging the roles of space and time, evolving the system using a space transfer matrix rather than the time evolution operator.The infinite-volume limit is then described by the fixed points of the latter transfer matrix, also known as influence matrices.To establish the potential of this method as a bona fide computational scheme, it is important to understand whether the influence matrices can be efficiently encoded in a classical computer.Here we begin this quest by presenting a systematic characterization of their entanglement-dubbed temporal entanglement-in chaotic quantum systems.We consider the most general form of space evolution, i.e., evolution in a generic spacelike direction, and present two fundamental results.First, we show that temporal entanglement always follows a volume law in time.Second, we identify two marginal cases-(i) pure space evolution in generic chaotic systems and (ii) any spacelike evolution in dual-unitary circuits-where Rényi entropies with index larger than one are sublinear in time while the von Neumann entanglement entropy grows linearly.We attribute this behavior to the existence of a product state with large overlap with the influence matrices.This unexpected structure in the temporal entanglement spectrum might be the key to an efficient computational implementation of the space evolution.
The two-state-vector formalism and the entangled histories formalism are attempts to better understand quantum correlations in time. Both formalisms share some similarities, but they are not identical, having subtle differences in their interpretation and manipulation of quantum temporal structures. However, the main objective of this paper is to prove that, with appropriately defined scalar products, both formalisms can be made isomorphic. We show, for instance, that they treat operators and states on equal footing, leading to the same statistics for all measurements. In particular, we discuss the topic of quantum correlations in time and show how they can be generated and analyzed in a consistent way using these formalisms. Furthermore, we elaborate on an unconventional behavior of quantum histories of evolving multipartite systems which do not exhibit global nonlocal correlations in time but nevertheless can lead to entangled reduced histories characterizing evolution of an arbitrarily chosen subsystem.
In this work, a canonical method to compute entanglement entropy is proposed. We show that for two-dimensional conformal theories defined in a torus, a choice of moduli space allows the typical entropy operator of the TFD to provide the entanglement entropy of the degrees of freedom defined in a segment and their complement. In this procedure, it is not necessary to make an analytic continuation from the Rényi entropy and the von Neumann entanglement entropy is calculated directly from the expected value of an entanglement entropy operator. We also propose a model for the evolution of the entanglement entropy and show that it grows linearly with time.
Abstract Entropy has played an essential role in the history of physics. Its mathematical definition and applications have changed over time till today. In this paper, we first review the historical evolution of these various points of view, from the thermodynamic definition to information entropy from Shannon in classical physics, up to the modern concept of Neumann’s quantum entropy. As a specific example, we consider entanglement entropy and compare the phase space approach in classical physics to the Hilbert space approach in quantum physics in simple model systems. We derive a general expression for the entanglement entropy of fermions and bosons in arbitrary partitions of Hilbert space, valid beyond the thermodynamic limit. Next, we compare thermodynamic heat engines with quantum heat engines. Finally, we proceed to the more general concept of quantum (computational) complexity and argue, using the concept of entanglement entropy, that the Heisenberg time in classically chaotic systems coincides with the time when maximal complexity is reached in the quantum case for systems with all–all interactions.
In this paper we state a fundamental question about the structure of correlations in time and analyze temporal monogamy relations. We show that the nature of temporal correlations is inherently different from the spatial ones but in similarity to quantum spatial correlations, we expose a phenomenon of monogamy of quantum entanglement in time. We perform this task applying the entangled histories framework as a modifcation of the consistent histories approach. These considerations are supported by introduction of necessary tools specific for the tensor algebra used for representation of spatial correlations. We show that Tsirelson bound on temporal Bell-like inequalities can be derived from the entangled histories approach. Finally, we point out that in a context of the tensor algebra used for linking states in different times further studies on mathematical structure of the state representing evolving systems are needed.