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arXiv:quant-ph/9706001 · Journal of Mathematical Physics

On tracial operator representations of quantum decoherence functionals

Oliver Rudolph, J. D. Maitland Wright

Published Nov 1, 1997• 11 citations
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Abstract

A general quantum history theory can be characterized by the space of histories and by the space of decoherence functionals. In this note we consider the situation where the space of histories is given by the lattice of projection operators on an infinite dimensional Hilbert space H. We study operator representations for decoherence functionals on this space of histories. We first give necessary and sufficient conditions for a decoherence functional being representable by a trace class operator on H⊗H, an infinite dimensional analogue of the Isham–Linden–Schreckenberg representation for finite dimensions. Since this excludes many decoherence functionals of physical interest, we then identify the large and physically important class of decoherence functionals which can be represented, canonically, by bounded operators on H⊗H.

DOI
10.1063/1.532157

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A general quantum history theory can be characterized by the space of histories and by the space of decoherence functionals. In this note we consider the situation where the space of histories is given by the lattice of projection operators on an infinite dimensional Hilbert space H. We study operator representations for decoherence functionals on this space of histories. We first give necessary and sufficient conditions for a decoherence functional being representable by a trace class operator on H⊗H, an infinite dimensional analogue of the Isham–Linden–Schreckenberg representation for finite dimensions. Since this excludes many decoherence functionals of physical interest, we then identify the large and physically important class of decoherence functionals which can be represented, canonically, by bounded operators on H⊗H.
Gell-Mann and Hartle have proposed a significant generalization of quantum theory in which decoherence functionals perform a key role. Isham–Linden–Schreckenberg have given a penetrating analysis of decoherence functionals for L(H), where H is finite dimensional (dimension greater than 2). In this note their conjecture on the significance of boundedness is verified. In particular, it is shown that when d is a bounded decoherence functional associated with a von Neumann algebra A, then, provided A has no direct summand of type I2, d can be represented as the difference between semi-innerproducts on A.
Recently, Schreckenberg investigated symmetries in the context of history quantum theories. In the case that the space of histories is given by the set of projectors on some finite dimensional Hilbert space he obtained, first, a complete characterization of all physical symmetries of history quantum theories — an analog of Wigner’s theorem — and, second, a complete mathematical characterization of symmetries of single decoherence functionals. In this paper we extend Schreckenberg’s results to the case where the underlying space of histories is given by the set of projectors on some infinite dimensional Hilbert space.
Gell-Mann and Hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are ‘‘histories’’ and decoherence functionals. Within this scheme it is natural to identify the space 𝒰𝒫 of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where 𝒰𝒫 is the lattice of projectors 𝒫(𝒱) in some Hilbert space 𝒱; in effect, one seeks the history analog of Gleason’s famous theorem in standard quantum theory. In the present article the solution to this problem for the case where 𝒱 is finite dimensional is presented. In particular, it is shown that every decoherence functional d(α,β), α,β∈𝒫(𝒱) can be written in the form d(α,β)=tr𝒱⊗𝒱(α⊗βX) for some operator X on the tensor-product space 𝒱⊗𝒱.
The basic ingredients of the `consistent histories' approach to a generalized quantum theory are `histories'and decoherence functionals. The main aim of this program is to find and to study the behaviour of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham it is natural to identify the space \\UP of propositions about histories with an orthoalgebra or lattice. When \\UP is given by the lattice of projectors \\PV in some Hilbert space \\V, consistent sets correspond to certain partitions of the unit operator in \\V into mutually orthogonal projectors \\{\\a_1,\\a_2,\\ldots\\}, such that the function d(\\a,\\a) is a probability distribution on the boolean algebra generated by \\{\\a_1,\\a_2,\\ldots\\}. Using the classification theorem for decoherence functionals, proven previously, we show that in the case where \\V is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(\\a) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(\\a,\\a)=p(\\a) holds.
The basic ingredients of the ‘‘consistent histories’’ approach to a generalized quantum theory are ‘‘histories’’ and decoherence functionals. The main aim of this program is to find and to study the behavior of consistent sets associated with a particular decoherence functional d. In its recent formulation by Isham [‘‘Quantum logic and the histories approach to quantum theory,’’ J. Math. Phys. 35, 2157–2185 (1994)] it is natural to identify the space 𝒰𝒫 of propositions about histories with an orthoalgebra or lattice. When 𝒰𝒫 is given by the lattice of projectors 𝒫(𝒱) in some Hilbert space 𝒱, consistent sets correspond to certain partitions of the unit operator in 𝒱 into mutually orthogonal projectors {α1,α2,...}, such that the function d(α,α) is a probability distribution on the boolean algebra generated by {α1,α2,...}. Using the classification theorem for decoherence functionals proven in Isham et al. [‘‘The classification of decoherence functionals: An analog of Gleason’s Theorem,’’ J. Math. Phys. 35, 6360–6370 (1994)] we show that in the case where 𝒱 is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution p(α) on the corresponding boolean algebra, decoherence functionals d with respect to which this set is consistent and which are such that for the probability functions d(α,α)=p(α) holds.