The nonlocality of quantum states on a bipartite system A+B is testedby comparing probabilistic outcomes of two local observables of differentsubsystems. For a fixed observable A of the subsystem A, its optimalapproximate double A' of the other system B is defined such that theprobabilistic outcomes of A' are almost similar to those of the fixed observableA. The case of a -finite standard von Neumann algebras is considered and theoptimal approximate double A' of an observable A is explicitly determined.The connection between optimal approximate doubles and quantumcorrelations is explained. Inspired by quantum states with perfect correlation,like Einstein–Podolsky–Rosen states and Bohm states, the nonlocality power ofan observable A for general quantum states is defined as the similarity that theoutcomes of A look like the properties of the subsystem B corresponding to A'.As an application of optimal approximate doubles, maximal Bell correlationof a pure entangled state on B(C~2)
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