On the concept of EPR states and their structure

Details On the concept of EPR states and their structure On the concept of EPR states and their structure

1999-10-14

arxiv.org/abs/quant-ph/9910066v2

Journal of Mathematical Physics, vol. 41, n. 2 (2000) 638.

Physics

Quantum Physics

Plain TeX, 22 pages

Scientific paper

10.1063/1.533156

In this paper the notion of an EPR state for the composite S of two quantum systems S1, S2, relative to S2 and a set O of bounded observables of S2, is introduced in the spirit of classical examples of Einstein-Podolsky-Rosen and Bohm. We restrict ourselves mostly to EPR states of finite norm. The main results are contained in Theorem 3,4,5,6 in section III and imply that if the EPR states of finite norm relative to (S2, O) exist, then the elements of O have discrete probability distributions and the Von Neuman algebra generated by O is essentially inbeddable inside S1 by an antiunitary map. The EPR states then correspond to the different imbeddings and certain additional parameters, and are explicitely given by formulae which generalize the famous example of Bohm. If O generates all bounded observables, S2 must be of finite dimension and can be imbedded inside S1 by an antiunitary map, and the EPR states relative to S2 are then in canonical bijection with the different imbeddings of S2 inside S1; moreover they are given by formulae which are exactly those of the generalized Bohm states. The notion of EPR states of infinite norm is also explored and it is shown that the original state of Einstein-Podolsky-Rosen can be realized as a renormalized limit of EPR states of finite quantum systems considered by Weyl, Schwinger and many others. Finally, a family of states of infinite norm generalizing the Einstein-Podolsky-Rosen example is explicitly given.

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