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Bipartite quantum entangled systems can exhibit measurement correlations thatviolate Bell inequalities, revealing the profoundly counter-intuitive nature ofthe physical universe. These correlations reflect the impossibility ofconstructing a joint probability distribution for all values of all thedifferent properties observed in Bell inequality tests. Physically, theimpossibility of measuring such a distribution experimentally, as a set ofrelative frequencies, is due to the quantum back-action of projectivemeasurements. Weakly coupling to a quantum probe, however, produces minimalback-action, and so enables a weak measurement of the projector of oneobservable, followed by a projective measurement of a non-commuting observable.By this technique it is possible to empirically measure weak-valuedprobabilities 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 ofthe Bell inequality, which we independently measure. This is possible becauseour distribution, like the weak values for projectors on which it is built, isnot constrained to the interval [0, 1]. It was first pointed out by Feynmanthat, for explaining singlet-state correlations within "a [local] hiddenvariable view of nature ... everything works fine if we permit negativeprobabilities". However, there are infinitely many such theories. Our method,involving "weak-valued probabilities", singles out a unique set ofprobabilities, and moreover does so empirically.

Using weak values to experimentally determine "negative probabilities" in a two-photon state with Bell correlations