Hardy’s criterion of nonlocality for mixed states

Abstract

We generalize Hardy's proof of nonlocality to the case of bipartite mixed statistical operators, and we exhibit a necessary condition which has to be satisfied by any given mixed state σ in order that a local and realistic hidden variable model exists which accounts for the quantum mechanical predictions implied by σ. Failure of this condition will imply both the impossibility of any local explanation of certain joint probability distributions in terms of hidden variables and the nonseparability of the considered mixed statistical operator. Our result can be also used to determine the maximum amount of noise, arising from imperfect experimental implementations of the original Hardy's proof of nonlocality, in presence of which it is still possible to put into evidence the nonlocal features of certain mixed states.

Hardy’s test versus the Clauser-Horne-Shimony-Holt test of quantum nonlocality: Fundamental and practical aspects

“…The sets in question are sets of values of hidden variables which imply certain outcomes of measurements. Following the steps in [26], only slightly generalized to different ǫ ν and equalities instead of bounds for the joint probabilities, we immediately find the necessary condition…”

“…Hardy managed to construct a pure state for which, according to QM, just this happens [17]. Later, his argument was generalized and it was shown that almost any pure state of any system with an arbitrary number of particles and arbitrary dimension of Hilbert space can be used [30,31,32], and even a large class of mixed states [26].…”

“…The key to generalizing Hardy's argument to finite values of the ǫ i is replacing logical implications by set-theoretical inclusions. This approach was pioneered very recently by Ghirardi and Marinatto (GM) [26,27]. The sets in question are sets of values of hidden variables which imply certain outcomes of measurements.…”

“…Following the lines of Refs. [18,26,27,55] we implement the measurements X 1 and Y 1 by switching the beam splitter B 1 between the two modes associated with the unitary matrices U 1 = U B (π/4) and W 1 = U P (2φ 0 )U B (π/4)U P (−2φ 0 ), and the beam splitter B 2 between the two modes U 2 = U B (0) and…”

“…We discuss the relation between Hardy's test and the CHSH test both in the absence and presence of imperfections. In the former case, we study the relation between the two tests in geometrical terms, making obvious the connection between the convex sets of joint probabilities of different measurements involved, while the latter is examined by means of set theoretical arguments [26,27,28]. We then propose a mesoscopic circuit which allows to implement both Hardy's test and the CHSH test by a simple change of parameters.…”