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Journal of Physics A Mathematical and General

Entropy and the central limit theorem in quantum mechanics

R. F. Streater

Published Sep 11, 1987• 8 citations
Publisher

Abstract

Gives a short proof that any infinitely divisible even representation of a Clifford algebra or a CAR algebra is quasi-free, and of Hudson's central limit theorem (1973) for fermions. The author shows that the entropy of any even state of the CAR, Clifford or CCR algebra is less than or equal to the entropy of the quasi-free state with the same two-point function.

DOI
10.1088/0305-4470/20/13/033

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Gives a short proof that any infinitely divisible even representation of a Clifford algebra or a CAR algebra is quasi-free, and of Hudson's central limit theorem (1973) for fermions. The author shows that the entropy of any even state of the CAR, Clifford or CCR algebra is less than or equal to the entropy of the quasi-free state with the same two-point function.
Abstract The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.
We study the entropy of pure shift-invariant states on a quantum spin chain. Unlike the classical case, the local restrictions to intervals of length N are typically mixed and have therefore a nonzero entropy SN which is, moreover, monotonically increasing in N. We are interested in the asymptotics of the total entropy. We investigate in detail a class of states derived from quasi-free states on a CAR algebra. These are characterized by a measurable subset of the unit interval. As the entropy density is known to vanish, SN is sublinear in N. For states corresponding to unions of finitely many intervals, SN is shown to grow slower than log2 N. Numerical calculations suggest a log N behavior. For the case with infinitely many intervals, we present a class of states for which the entropy SN increases as Nα where α can take any value in (0,1).
Quantum Field Theory (QFT) represents a vast generalization of Quantum Mechanics (QM), as it deals with systems that have an infinite number of degrees of freedom. The Stone-von Neumann theorem, which establishes the equivalence of irreducible representations of the canonical commutation relations (CCR) in QM, does not extend to QFT. Consequently, QFT admits multiple inequivalent irreducible representations, leading to a much richer algebraic structure. This essay aims to explore the physics of QFT from the operator algebra perspective, particularly focusing on entanglement entropy. We discuss the role of von Neumann algebras of different types in QFT, describe the local operator algebra approach to QFT, and explain how entanglement entropy can be defined in terms of the algebra of observables. Additionally, we explore the benefits of this approach in concrete applications, specifically in quantum field theory on curved spacetime.