Hilbert space separability and the Einstein-Podolsky-Rosen state

Quantum mechanics is formulated on a Hilbert space that is assumed to be separable. However, there seems to be no clear reason justifying this assumption. Does it have physical implications? We answer in the positive by proposing a test that witnesses the non-separability of the Hilbert space, at the expense of requiring measurements with uncountably many outcomes. In the search for a less elusive manifestation of non-separability, we consider the original Einstein-Podolsky-Rosen (EPR) state as a candidate for possessing nonlocal correlations stronger than any state in a separable Hilbert space. Nevertheless, we show that, under mild assumptions, this state is not a vector in any bipartite space, even non-separable, and therefore cannot be described within the standard Hilbert space formalism.