C*-Algèbres des systèmes canoniques. I

Loupias, G.; Miracle-Sole, Salvador

doi:10.1007/bf01773339

Cited by 77 publications

(58 citation statements)

References 9 publications

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“…must be positive semidefinite, which becomes the necessary condition forρ q to be positive semidefinite. According to KLM [14,15], the converse is also true, i.e., the positive semidefiniteness of all matrices {M ij } becomes sufficient for that ofρ q . In our heuristic argument, this sufficiency may be seen, with less mathematical rigor, as follows.…”

Section: Klm Conditionsmentioning

confidence: 99%

Complete conditions for legitimate Wigner distributions

Nha

2008

Phys. Rev. A

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Given a real-valued phase-space function, it is a nontrivial task to determine whether it corresponds to a Wigner distribution for a physically acceptable quantum state. This topic has been of fundamental interest for long, and in a modern application, it can be related to the problem of entanglement detection for multi-mode cases. In this paper, we present a hierarchy of complete conditions for a physically realizable Wigner distribution. Our derivation is based on the normally-ordered expansion, in terms of annihilation and creation operators, of the quasi-density operator corresponding to the phase-space function in question. As a by-product, it is shown that the phase-space distributions with elliptical symmetry can be readily diagonalized in our representation, facilitating the test of physical realizability. We also illustrate how the current formulation can be connected to the detection of bipartite entanglement for continuous variables.

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Section: Klm Conditionsmentioning

confidence: 99%

Complete conditions for legitimate Wigner distributions

Nha

2008

Phys. Rev. A

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“…By the parametrix formula in the scalar case, to the above q, there correspond a positive integer r, an element gEC co (W) 9 and an element heC q (W) such that Remark. One may find a prototype of the above theorem in G. Loupias and S. Miracle-Sole's paper [5], which tells us that any operator of the Schrodinger representation can be considered as a tempered distribution on the phase space of a system with n degrees of freedom.…”

Section: = Fl(s)g(t-s)dsmentioning

confidence: 99%

On a Fourier Expansion In Continuous Crossed Products

Takai

1975

Publ. Res. Inst. Math. Sci.

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Let (M, R, a) be a separable continuous PF*-dynamical system such that M is R-finite.Then any element in the crossed product R(g a M of M by a can be expressed as a vector valued tempered distribution D^T^ which. is a weak* limit of Tt£<=K(R; S3) in the dual space 5^(R; t))* of a generalized Schwartz space S r7 (R; tj), where A"(R; 93) is the Tomita algebra corresponding to R(x)«M. § 1. Introduction , using a duality for crossed products, established a structure theorem that any factor of type III with separable predual can be written as the crossed product of a von Neumann algebra of type IIa, by a continuous action of the real numbers. A. Connes [2] verified that there exist factors of type III which are isomorphic to no discrete crossed product of a semifinite von Neumann algebra by an Communicated by H. Araki, September 26, 1975.

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“…The function a(p', x') is the « symplectic Fourier transform » of a(x, p) [2]. It results from a special identification between the dual of R 2 " and R 2 " itself.…”

Section: The Weyl Correspondencementioning

confidence: 99%

“…We shall now exhibit the irreducible sector lo{h) (which is denoted by 3^ in [4] and [2]). Consider the gaussian To end this section, we mention how the unitary equivalences between, for instance, the representations (a), (?)…”

Section: If F and G Belong To L^rmentioning

confidence: 99%