On the concept of Einstein-Podolsky-Rosen states and their structure

In this paper the notion of an EPR state for the composite S of two quantum systems S-1,S-2, relative to S-2 and a set O of bounded observables of S-2, is introduced in the spirit of the classical examples of Einstein-Podolsky-Rosen and Bohm. We restrict ourselves mostly to EPR states of finite norm. The main results are contained in Theorems 3-6 and imply that if EPR states of finite norm relative to (S-2,O) exist, then the elements of O have discrete probability distributions and the Von Neumann algebra generated by O is essentially imbeddable inside S-1 by an antiunitary map. The EPR states then correspond to the different imbeddings and certain additional parameters, and are explicitly given by formulas which generalize the famous example of Bohm. If O generates all bounded observables, S-2 must be of finite dimension and can be imbedded inside S-1 by an antiunitary map, and the EPR states relative to S-2 are then in canonical bijection with the different imbeddings of S-2 inside S-1; moreover they are then given by formulas 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. (C) 2000 American Institute of Physics. [S0022-2488(00)02002-8].