  Different ensembles of the same density matrix are indistinguishable within
the modern Kolmogorov probability measure theory of quantum random phenomena.
We find that changing the framework from the Kolmogorov one to a
frequentist-inspired theory of quantum random phenomena -- a la von Mises --
would lift the indistinguishability, and potentially cost us the no-signaling
principle (i.e., lead to superluminal communication). We believe that this adds
to the recent works on the search for a suitable representation of the state of
a quantum system. While erstwhile arguments for potential modifications in the
representation of the quantum state were restricted to possible variations in
the formalism of the quantum theory, we indicate a possible fallout of altering
the underlying theory of random processes.
