Extended Boole-Bell Inequalities Applicable to Quantum Theory

Abstract It is shown by using a rather elementary argument in Mathematical Logic that if indeed, quantum theory does violate the famous Bell Inequalities, then quantum theory must inevitably also violate all valid mathematical statements, and in particular, such basic algebraic relations like 0 = 0, 1 = 1, 2 = 2, 3 = 3, … and so on … An interest in that result is due to the following three alternatives which it imposes upon both Physics and Mathematics: Quantum Theory is inconsistent. Quantum Theory together with Mathematics are inconsistent. Mathematics is inconsistent. In this regard one should recall that, up until now, it is not known whether Mathematics is indeed consistent.

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Bell’s theorem

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-q004-1 ◽

2018 ◽

Author(s):

Arthur Fine

Keyword(s):

Quantum Mechanics ◽

Quantum Theory ◽

Experimental Tests ◽

Bell Inequalities ◽

Bell’S Theorem ◽

Bell's Theorem ◽

Bell Theorem ◽

System Of Inequalities ◽

Action At A Distance ◽

High Degree

Bell’s theorem is concerned with the outcomes of a special type of ‘correlation experiment’ in quantum mechanics. It shows that under certain conditions these outcomes would be restricted by a system of inequalities (the ‘Bell inequalities’) that contradict the predictions of quantum mechanics. Various experimental tests confirm the quantum predictions to a high degree and hence violate the Bell inequalities. Although these tests contain loopholes due to experimental inefficiencies, they do suggest that the assumptions behind the Bell inequalities are incompatible not only with quantum theory but also with nature. A central assumption used to derive the Bell inequalities is a species of no-action-at-a-distance, called ‘locality’: roughly, that the outcomes in one wing of the experiment cannot immediately be affected by measurements performed in another wing (spatially distant from the first). For this reason the Bell theorem is sometimes cited as showing that locality is incompatible with the quantum theory, and the experimental tests as demonstrating that nature is nonlocal. These claims have been contested.

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Is a time symmetric interpretation of quantum theory possible without retrocausality?

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0607 ◽

2017 ◽

Vol 473 (2202) ◽

pp. 20160607 ◽

Cited By ~ 30

Author(s):

Matthew S. Leifer ◽

Matthew F. Pusey

Keyword(s):

Quantum Theory ◽

Quantum State ◽

General Framework ◽

Bell Inequalities ◽

Time Symmetry ◽

Causality Criterion ◽

The Status ◽

Interpretation Of Quantum Theory ◽

Huw Price ◽

New Interpretation

Huw Price has proposed an argument that suggests a time symmetric ontology for quantum theory must necessarily be retrocausal, i.e. it must involve influences that travel backwards in time. One of Price's assumptions is that the quantum state is a state of reality. However, one of the reasons for exploring retrocausality is that it offers the potential for evading the consequences of no-go theorems, including recent proofs of the reality of the quantum state. Here, we show that this assumption can be replaced by a different assumption, called λ -mediation, that plausibly holds independently of the status of the quantum state. We also reformulate the other assumptions behind the argument to place them in a more general framework and pin down the notion of time symmetry involved more precisely. We show that our assumptions imply a timelike analogue of Bell's local causality criterion and, in doing so, give a new interpretation of timelike violations of Bell inequalities. Namely, they show the impossibility of a (non-retrocausal) time symmetric ontology.

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NONLOCALITY IMPROVES DEUTSCH ALGORITHM

International Journal of Quantum Information ◽

10.1142/s0219749909005110 ◽

2009 ◽

Vol 07 (03) ◽

pp. 603-614

Author(s):

KOJI NAGATA ◽

SANGKYUNG LEE ◽

JAEWOOK AHN

Keyword(s):

Hilbert Space ◽

Quantum Theory ◽

Unknown Function ◽

Quantum Computer ◽

Bell Inequalities ◽

Speed Up ◽

New Type ◽

Space Formalism ◽

Entangled Photon Pairs ◽

Classical Computer

We show that the Bell inequalities lead to a new type of linear-optical Deutsch algorithms. We have considered the use of entangled photon pairs to determine probabilistically two unknown functions. The usual Deutsch algorithm determines one unknown function and exhibits a two to one speed up in a certain computation on a quantum computer rather than on a classical computer. We found that the violation of Bell locality in the Hilbert space formalism of quantum theory predicts that the proposed probabilistic Deutsch algorithm for computing two unknown functions exhibits at least a [Formula: see text] to one speed up.

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Causation and Locality

10.1093/oso/9780198714057.003.0010 ◽

2017 ◽

Author(s):

Richard Healey

Keyword(s):

Quantum Theory ◽

Causal Explanation ◽

Bell Inequalities ◽

Space Time ◽

Counterfactual Dependence ◽

The Other ◽

Entangled State ◽

Space Time Structure ◽

Action At A Distance ◽

Time Point

By moving to the context of relativistic space-time structure, this chapter completes the argument of Chapter 4 that we can use quantum theory locally to explain correlations that violate Bell inequalities with no instantaneous action at a distance. Chance here must be relativized not just to time but to a space-time point, so that an event may have more than one chance at the same time—it may even be certain relative to one space-time point but ‘at the same time’ completely uncertain relative to another. This renders Bell’s principle of Local Causality either inapplicable or intuitively unmotivated. Counterfactual dependence between the outcomes of measurements on systems assigned an entangled state is not causal since neither outcome is subject to intervention: but it may still be appealed to in a non-causal explanation of one in terms of the other.

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Turing, ciphers and quanta

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2011.0324 ◽

2012 ◽

Vol 370 (1971) ◽

pp. 3418-3431 ◽

Cited By ~ 1

Author(s):

Artur Ekert ◽

Alastair Kay ◽

James Pope

Keyword(s):

Quantum Theory ◽

Quantum Cryptography ◽

Quantum Entanglement ◽

Secure Communication ◽

Bell Inequality ◽

Bell Inequalities ◽

Foundations Of Quantum Theory ◽

The Past ◽

Alan Turing ◽

Widespread Belief

Alan Turing has certainly contributed to a widespread belief that the quest for a perfect, unbreakable, cipher is a futile pursuit. The ancient art of concealing information has, in the past, been matched by the ingenuity of code-breakers, but no longer! With the advent of quantum cryptography, the hopes of would-be eavesdroppers have been dashed, perhaps for good. Moreover, recent research, building on schemes that were invented decades ago to perform quantum cryptography, shows that secure communication certified by a sufficient violation of a Bell inequality makes a seemingly insane scenario possible—devices of unknown or dubious provenance, even those that are manufactured by our enemies, can be safely used for secure communication, including key distribution. All that is needed to implement this bizarre and powerful form of cryptography is a loophole-free test of a Bell inequality, which is on the cusp of technological feasibility. We provide a brief overview of the intriguing connections between Bell inequalities and cryptography and describe how studies of quantum entanglement and the foundations of quantum theory influence the way we may protect information in the future.

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VIOLATION OF CLASSICAL INEQUALITIES AND EPR CORRELATIONS IN A TWO-MODE THREE-LEVEL ATOMIC SYSTEM

International Journal of Quantum Information ◽

10.1142/s0219749908003761 ◽

2008 ◽

Vol 06 (04) ◽

pp. 885-898 ◽

Cited By ~ 2

Author(s):

EYOB A. SETE

Keyword(s):

Quantum Theory ◽

Atomic System ◽

Bell Inequality ◽

Bell Inequalities ◽

Schwarz Inequality ◽

Atomic Coherence ◽

Cascade System ◽

Epr Correlations

We investigate violation of Cauchy–Schwarz and Bell inequalities in two-mode three-level cascade system with injected atomic coherence in the framework of quantum theory of multiwave mixing. We show that Cauchy–Schwarz inequality is strongly violated when there is strong entanglement in the system. It also appears that Bell inequality is violated in the region where there is weak entanglement and well-preserved where there is strong entanglement in the system. We thus note that there are states which are entangled but do not violate Bell inequality. We also show that this system can be used to prepare states that exhibit EPR correlations.

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