  Bipartite quantum entangled systems can exhibit measurement correlations that
violate Bell inequalities, revealing the profoundly counter-intuitive nature of
the physical universe. These correlations reflect the impossibility of
constructing a joint probability distribution for all values of all the
different properties observed in Bell inequality tests. Physically, the
impossibility of measuring such a distribution experimentally, as a set of
relative frequencies, is due to the quantum back-action of projective
measurements. Weakly coupling to a quantum probe, however, produces minimal
back-action, and so enables a weak measurement of the projector of one
observable, followed by a projective measurement of a non-commuting observable.
By this technique it is possible to empirically measure weak-valued
probabilities for all of the values of the observables relevant to a Bell test.
The marginals of this joint distribution, which we experimentally determine,
reproduces all of the observable quantum statistics including a violation of
the Bell inequality, which we independently measure. This is possible because
our distribution, like the weak values for projectors on which it is built, is
not constrained to the interval [0, 1]. It was first pointed out by Feynman
that, for explaining singlet-state correlations within "a [local] hidden
variable view of nature ... everything works fine if we permit negative
probabilities". However, there are infinitely many such theories. Our method,
involving "weak-valued probabilities", singles out a unique set of
probabilities, and moreover does so empirically.
