Stronger violation of local theories with equalities

Abstract

Bell type inequalities are used to test local realism against quantum theory. In this paper, we consider a two party system with two settings and two possible outcomes on each side, and derive equalities in local theories which are violated by quantum theory by a factor of 1.522 tolerating 0.586 fraction of white noise admixture which is twice that of the previous results.November 8, 2018 I. INTRODUCTIONThe idea of local realism opened its way into quantum theory by the work of Einstein, Podolsky and Rosen which is well known as EPR paradox [1], though it is not really a paradox, because it has been argued recently that the assumptions of EPR were wrong [2]. However it was left to John S. Bell who derived an inequality based on local theories and proved that it was violated by statistical predictions of quantum theory [3,4]. Since then many attempts have been made to derive Bell type inequalities which are violated by a stronger factor, so that it can be tested in the real experiments in which errors are inevitable. Among these are Clauser, Horne, Shimony and Holt inequality, the so called CHSH inequality [5,6].Nowadays, with the growing power of computers, the numerical methods have attracted attentions for constructing these inequalities as much as possible [7-9] though analytical approaches are in progress too [10]; with the hope that some of which could be violated by a stronger factor.In this paper we first introduce a simple but general method for constructing Bell expressions which can be applied to two party experiments each measuring two observables with different outputs. Generalization to more parties and/or more observables is straightforward. Then in a special case of a two party system with two settings and two possible outcomes on each side, we use these expressions and show that according to local theories there exist equalities which are violated by quantum theory by a stronger factor than Bell type inequalities. Meanwhile we derive an inequality in this case which is violated by a factor of 1.621 tolerating 0.293 fraction of white noise admixture. Although in this case the tolerance is the same as those derived previously in the literature the range of violation is greater. II. DERIVATION OF BELL TYPE EXPRESSIONSLet's consider a two party system with no known interaction between its parts. Suppose the left party, say A performs two possible measurements a with outcomes i ∈ {0, · · · , m − 1} and a ′ with outcomes i ′ ∈ {0, · · · , m ′ − 1}. Similarly the right party, say B, performs two possible measurements b with outcomes j ∈ {0, · · · , n − 1} and b ′ with outcomes j ′ ∈ {0, · · · , n ′ − 1}. For simplicity, from now on we use unprimed/primed variables and indexes for the first/second measurement for each party whenever applicable and label such a system as mn ⊗ m ′ n ′ . As a consequence of locality, for measurements which are not simultaneous, there exists a probability q ii ′ jj ′ aa ′ bb ′ defined as the probability that measurements of a results i, a ′ results i ′ , b results j and b ′ r...

Violation of Equalities in Bipartite Qutrits Systems

“…As has been proved in [17], two group of constraints, i.e. normalizability of joint probabilities and no-signaling reduces the number of independent P 's to…”

“…We have fully discussed this in [17]. Also following the discussion in [17], in local theories a Bell expression can be written as…”

“…The reason is that one of the above constraints can be written in terms of the others and incidentally, this is not taken into account in [18]. We have fully discussed this in [17]. Also following the discussion in [17], in local theories a Bell expression can be written as…”

“…( 20), is 0.50203 which exceeds that of previous results by 0.19818 and with respect to inequality ( 14) the tolerance is increased by 0.18817. Note that in the presence of white noise the value of the Bell expression (17), denoted by E noisy c (p), would be…”

“…Recently based on numerical calculations it has been shown in [17] that in 22⊗22 case, there exist Bell expressions in local theories with exact value, which we called them equalities, and has been proved that these equalities are violated more strongly than inequalities. In fact in our equalities the amount of violation and the tolerance of white noise admixture are 0.41421 and 0.58579 respectively.…”