Comment on "Resolution of the Einstein-Podolsky-Rosen and Bell Paradoxes"

Abstract

In a recent letter, 1 Pitowsky has given a model of electron spin in which "Every electron at each given moment has a definite spin in all directions", but which, he claims, does not imply Bell's inequality. A non-Kolmogorov probability theory in the model prevents the usual proofs of Bell's inequality from going through. I give here a very simple proof of a Bell-type inequality from the quoted statement. The inequality shows that the statement is inconsistent with quantum mechanics.Consider N pairs of electrons in the singlet state. One member of each pair moves to the left and the other to the right. Let N (A + : C + ) be the number of pairs in which the left member has spin up in the A direction and the right member has spin up in the C direction. Let N (A + C − :) be the number in which the left member has spin up in the A direction and spin down in the C direction. According to the quoted statement, these are meaningful quantities. ThenQuantum mechanics predicts that if N (A + : C + ) is measured, then, where θ AC is the angle between A and C. According to the quoted statement N (A + : C + ) exists independently of whether it is measured or not and so the approximation holds whether it is measured or not. The above inequality is inconsistent with the approximation for θ AB = θ BC = 60 • and θ AC = 120 • 1 I. Pitowsky, Phys. Rev. Lett. 48, 1299Lett. 48, (1982.

Geometry of joint reality: Device-independent steering and operational completeness

“…Hence, since this rectangle corresponds to the value c = a·b in Theorem 1, we have (a · b) > 0 in Eq. (10).…”

“…Note that the same symbol, (c), has been used in Eqs. (10) and (24) for notational convenience, corresponding to formally equating the labels E 1 , E 2 , E 3 , E 4 with E + , E − , E + , E − , respectively. Note also that violation of inequality (24) requires an operational plane that extends outside all four sides of the dashed rectangle in Fig.…”

“…In particular, our results allow for scenarios in which the individual relative frequencies of random variables converge (to corresponding observable probabilities) as the size of the ensemble increases, but where their joint relative frequency does not converge (see, e.g., [9] for an example). This is analogous to strong proofs of Bell inequalities in which only finite counting arguments are required [9][10][11], thus allowing, for example, local hidden variable models based on properties of nonmeasurable sets [12] to be ruled out [10].…”

On the role of joint probability distributions of incompatible observables in Bell and Kochen–Specker Theorems

“…Finite sampling arguments to rule out realism appeared in the literature time ago [7]. Mermin [8] and Macdonald [9] employed some inequalities based on relative frequencies to discard the Pitowsky's hidden variable model in [21]. In such a model, equations such as (2) are formulated for compatible observables, e. g. a pair of spin components for Alice and Bob.…”

“…and define two sets: (9) so that N = A ∪ B. Then, suppose an ensemble of varying size N , such that for N ∈ A, (x, y) is a pair of random variables which samples the probability P A (x, y), and, for N ∈ B, (x, y) samples the probability P B (x, y).…”