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We present in this paper that the CHSH game admits one and only one optimalstate and so remove all ambiguity of representations. More precisely, we usethe well-known universal description of quantum commuting correlations as statespace on the universal algebra for two player games, and so allows us tounambigiously compare quantum strategies as states on this common algebra. Assuch we find that the CHSH game leaves a single optimal state on this commonalgebra. In turn passing to any non-minimal Stinespring dilation for thisunique optimal state is the only source of ambiguity (including self-testing):More precisely, any state on some operator algebra may be uniquely broken upinto its minimal Stinespring dilation as an honest representation for theoperator algebra followed by its vector state. Any other Stinespring dilationhowever arises simply as an extension of the minimal Stinespring dilation(i.e., as an embedding of the minimal Hilbert space into some random ambientone). As such this manifests the only source of ambiguity appearing in most(but not all!) traditional self-testing results such as for the CHSH game aswell as in plenty of similar examples. We then further demonstrate thesimplicity of our arguments on the Mermin--Peres magic square and magicpentagram game.  Most importantly however, we present this article as an illustration ofoperator algebraic techniques on optimal states and their quotients, and wefurther pick up the results of the current article in another following one(currently under preparation) to derive a first robust self-testing result inthe quantum commuting model.

The quantum commuting model (Ia): The CHSH game and other examples: Uniqueness of optimal states