Exclusivity principle forbids sets of correlations larger than the quantum set

Amaral, Bárbara; Cunha, Marcelo Terra; Cabello, Adán

doi:10.1103/physreva.89.030101

Cited by 43 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Holik

Bosyk

Bellomo

2015

Entropy

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In this article, we discuss the formal structure of a generalized information theory based on the extension of the probability calculus of Kolmogorov to a (possibly) non-commutative setting. By studying this framework, we argue that quantum information can be considered as a particular case of a huge family of non-commutative extensions of its classical counterpart. In any conceivable information theory, the possibility of dealing with different kinds of information measures plays a key role. Here, we generalize a notion of state spectrum, allowing us to introduce a majorization relation and a new family of generalized entropic measures.

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Section: Introductionmentioning

confidence: 99%

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

Holik

Bosyk

Bellomo

2015

Entropy

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“…The problem of whether or not the E principle (complemented by some assumptions) can explain all the quantum limits of the correlations between the outcomes of comeasurable sharp measurements is still open. However, taking into account the earlier successful predictions of the E principle [18][19][20][21][22][23] and the evidences of failure of other approaches [8][9][10], the result presented here promotes the E principle as the best option for understanding the limits of quantum contextual and nonlocal correlations. This apparent fundamental role of the E principle in QT is also supported by some recent results aiming to understand QT from fundamental physical principles [15,27,28].…”

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confidence: 99%

“…[22]) that the maximum quantum violation of another noncontextuality inequality is also singled out by the E principle (under the assumptions mentioned) [30].…”

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“…So far, the E principle, plus some assumptions, has been proven to single out the maximum quantum violation of the simplest noncontextuality inequality [18], to exclude all extremal nonlocal boxes in the (3, 2, 2) scenario [21,23], and to exclude any set of correlations strictly larger than the set of correlations achievable with quantum systems for any selfcomplementary graph of exclusivity [22]. There is also strong evidence that the E principle singles out the maximum quantum violation of all basic noncontextuality inequalities [20].…”

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“…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…”

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Exclusivity principle and the quantum bound of the Bell inequality

Cabello

2014

Phys. Rev. A

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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

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Introduction

Amaral¹,

Cunha²

2018

SpringerBriefs in Mathematics

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Exclusivity principle forbids sets of correlations larger than the quantum set

Cited by 43 publications

References 47 publications

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

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

Exclusivity principle and the quantum bound of the Bell inequality

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