  We present in this paper that the CHSH game admits one and only one optimal
state and so remove all ambiguity of representations. More precisely, we use
the well-known universal description of quantum commuting correlations as state
space on the universal algebra for two player games, and so allows us to
unambigiously compare quantum strategies as states on this common algebra. As
such we find that the CHSH game leaves a single optimal state on this common
algebra. In turn passing to any non-minimal Stinespring dilation for this
unique optimal state is the only source of ambiguity (including self-testing):
More precisely, any state on some operator algebra may be uniquely broken up
into its minimal Stinespring dilation as an honest representation for the
operator algebra followed by its vector state. Any other Stinespring dilation
however arises simply as an extension of the minimal Stinespring dilation
(i.e., as an embedding of the minimal Hilbert space into some random ambient
one). 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 as
well as in plenty of similar examples. We then further demonstrate the
simplicity of our arguments on the Mermin--Peres magic square and magic
pentagram game.
  Most importantly however, we present this article as an illustration of
operator algebraic techniques on optimal states and their quotients, and we
further pick up the results of the current article in another following one
(currently under preparation) to derive a first robust self-testing result in
the quantum commuting model.
