Exclusivity principle and the quantum bound of the Bell inequality

Abstract

We show that, for general probabilistic theories admitting sharp measurements, the exclusivity principle together with two assumptions exactly singles out the Tsirelson bound of the Clauser-Horne-Shimony-Holt Bell inequality.Introduction.-Quantum theory (QT) is arguably the most accurate scientific theory of all times. Nevertheless, despite its mathematical simplicity, its fundamental principles are still unknown. In the quest for these principles, a key question is why QT is exactly as nonlocal [1] and contextual [2] as it is.There are two different approaches to answer this question. On one hand, the "black box" approach, which focuses on correlations among the outcomes of measurements in multipartite scenarios. This approach makes no assumptions on the internal working of the measurement devices (which are treated as black boxes) and pays no attention to the postmeasurement state of the system (since measurements are treated as if they were demolition measurements). Within this approach, it has been proven that the principles of information causality [3] and macroscopic locality [4], complemented by some assumptions, single out the maximum quantum violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [5,6] (i.e., the Tsirelson bound [7]). However, it has also been proven that these principles cannot exclude extremal nonlocal boxes prohibited in QT in the three-party, two-outcome, two-setting or (3, 2, 2) scenario [8,9]. Moreover, by its very definition, the black box approach cannot explain quantum contextual correlations [2]. In addition, there is increasing evidence that the black box approach, with independence of the principle invoked, cannot explain even quantum nonlocal correlations [10].On the other hand, the "sharp measurements," "graphtheoretic," or "contextuality" approach [11][12][13] focus on correlations among the outcomes of sharp measurements. For general probabilistic theories (GPTs), sharp measurements are nondemolition measurements that are repeatable and cause minimal disturbance [14,15]. In QT, sharp measurements are projective measurements [16]. In QT, generalized measurements are represented by positive operator-valued measures. However, in QT, any generalized measurement can always be implemented as a sharp measurement on the system and the environment [17].Within the sharp measurements approach, an "event" is defined as the postmeasurement state of the system after a sharp measurement. Two events are equivalent when they correspond to indistinguishable states. Two events are exclusive when there is a sharp measurement that perfectly distinguishes between them.The exclusivity (E) principle [18][19][20][21][22][23] states that any set of m pairwise exclusive events is m-wise exclusive. According to Kolmogorov's axioms of probability, the sum of the

Contextuality and Indistinguishability

“…Thus, there is no contradiction in assigning the value V = 1 to P i,j,k,l in proposition P 1 and the value 0 to proposition P 2 . This modification of the propositional structure and the truth values assignment in the quantum formalism, allows us to avoid the contradiction in Equations (3) and (12). In other words, the description of propositions using quasiset theory allows us to assign definite values to particles, but in a way that is very compatible with the constrains imposed by the quantum formalism.…”

The Unspeakable Why

“…This result is an improved version of an argument introduced in Ref. [21]. A similar argument allows us to derive the quantum limits for n-partite Bell-like inequalities for non-local (but not genuinely n-partite non-local) hidden variable theories [22].…”

“…As explained in Table I, there are 16 sets like this one. For each of them, there is a inequality like (21). If we sum all of them we obtain,…”

Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory

“…In particular, it has given rise to a quest for the foundational principles that singularize quantum mechanics among a vast family of possible statistical theories [2][3][4][5]. The study and characterization of quantum correlations play a central role in this quest [6], entanglement [7][8][9] and discord [10] being the most important ones.…”