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2012 | OriginalPaper | Buchkapitel

Bell's Inequalities — Foundations and Quantum Communication

verfasst von : Časlav Brukner, Marek Żukowski

Erschienen in: Handbook of Natural Computing

Verlag: Springer Berlin Heidelberg

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Abstract

For individual events, quantum mechanics makes only probabilistic predictions. Can one go beyond quantum mechanics in this respect? This question has been a subject of debate and research since the early days of the theory. Efforts to construct a deeper, realistic level of physical description, in which individual systems have, like in classical physics, preexisting properties revealed by measurements are known as hidden-variable programs. Demonstrations that a hidden-variable program necessarily requires outcomes of certain experiments to disagree with the predictions of quantum theory are called “no-go theorems.” The Bell theorem excludes local hidden variable theories. The Kochen–Specker theorem excludes non-contextual hidden variable theories. In local hidden-variable theories faster-than-light-influences are forbidden, thus the results for a given measurement (actual, or just potentially possible) are independent of the settings of other measurement devices which are at space-like separation. In non-contextual hidden-variable theories, the predetermined results of a (degenerate) observable are independent of any other observables that are measured jointly with it.

It is a fundamental doctrine of quantum information science that quantum communication and quantum computation outperform their classical counterparts. If this is to be true, some fundamental quantum characteristics must be behind the better-than-classical performance of information-processing tasks. This chapter aims at establishing connections between certain quantum information protocols and foundational issues in quantum theory. After a brief discussion of the most common misinterpretations of Bell's theorem and a discussion of what its real meaning is, we will demonstrate how quantum contextuality and violation of local realism can be used as useful resources in quantum information applications. In any case, the readers should bear in mind that this chapter is not a review of the literature of the subject, but rather a quick introduction to it.

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Vorheriges Kapitel Mathematics for Quantum Information Processing

Nächstes Kapitel Algorithms for Quantum Computers

Acín A, Scarani V, Wolf MM (2002) Bell inequalities and distillability in N-quantum-bit systems. Phys Rev A 66:042323MathSciNetCrossRef

Acín A, Gisin N, Masanes L (2006) From Bell's theorem to secure quantum key distribution. Phys Rev Lett 97:120405CrossRef

Aharon N, Vaidman L (2008) Quantum advantages in classically defined tasks. Phys Rev A 77:052310CrossRef

Ambainis A, Nayak A, Ta-Shma A, Vazirani U (1999) Dense quantum coding and a lower bound for 1-way quantum automata. Proceedings of the 31st annual ACM symposium on the theory of computing, New York, pp 376–383

Anders J, Browne DE (2009) Computational power of correlations. Phys Rev Lett 102:050502

Aspect A, Dalibard J, Roger G (1982) Experimental test of Bell's inequalities using time-varying analyzers. Phys Rev Lett 49:1804–1807MathSciNetCrossRef

Augusiak R, Horodecki P (2006) Bound entanglement maximally violating Bell inequalities: quantum entanglement is not fully equivalent to cryptographic security. Phys Rev A 74:010305MathSciNetCrossRef

Barrett J (2007) Information processing in generalized probabilistic theories. Phys Rev A 75:032304CrossRef

Bechmann-Pasquinucci H (2005) From quantum state targeting to Bell inequalities. Found Phys 35:1787–1804MathSciNetCrossRefMATH

Bell JS (1964) On the Einstein-Podolsky-Rosen paradox. Physics 1:195–200; reprinted in Bell JS (1987) Speakable and unspeakable in quantum mechanics. Cambridge University Press, Cambridge

Bell JS (1966) On the problem of hidden variables in quantum mechanics. Rev Mod Phys 38:447–452CrossRefMATH

Bell JS (1985) Free variables and local causality. Dialectica 39:103–106

Bennett CH, Wiesner SJ (1992) Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys Rev Lett 69:2881–2884MathSciNetCrossRefMATH

Bohm D (1952) A suggested interpretation of the quantum theory in terms of “hidden” variables, I and II. Phys Rev 85:166–193MathSciNetCrossRefMATH

Bohr N (1935) Can quantum-mechanical description of physical reality be considered complete? Phys Rev 48:696–702CrossRefMATH

Branciard C, Ling A, Gisin N et al. (2007) Experimental falsification of Leggett's nonlocal variable model. Phys Rev Lett 99:210407CrossRef

Brassard G (2003) Quantum communication complexity (a survey). Found Phys 33:1593–1616. Available via http://arxiv.org/abs/quant-ph/0101005

Brassard G, Buhrman H, Linden N et al. (2006) Limit on nonlocality in any world in which communication complexity is not trivial. Phys Rev Lett 96:250401MathSciNetCrossRef

Bremner M J, Mora C, Winter A (2009) Are random pure states useful for quantum computation? Phys Rev Lett 102:190502

Brennen GK, Miyake A (2008) Measurement-based quantum computer in the gapped ground state of a two-body Hamiltonian. Phys Rev Lett 101:010502CrossRef

Brukner Č, Paterek T, Żukowski M (2003) Quantum communication complexity protocols based on higher-dimensional entangled systems. Int J Quantum Inf 1:519–525CrossRefMATH

Brukner Č, Żukowski M, Zeilinger A (2002) Quantum communication complexity protocol with two entangled qutrits. Phys Rev Lett 89:197901CrossRef

Brukner Č, Taylor S, Cheung S, Vedral V (2004a) Quantum entanglement in time. Available via arXiv:quant-ph/0402127

Brukner Č, Żukowski M, Pan J-W, Zeilinger A (2004b) Bell's inequality and quantum communication complexity. Phys Rev Lett 92:127901MathSciNetCrossRef

Buhrman H, Cleve R, van Dam W (1997) Quantum entanglement and communication complexity. Available via http://arxiv.org/abs/quant-ph/9705033

Buhrman H, van Dam W, Høyer P, Tapp A (1999) Multiparty quantum communication complexity. Phys Rev A 60:2737–2741CrossRef

Cabello A, Estebaranz JM, García-Alcaine G (1996) Bell-Kochen-Specker theorem: a proof with 18 vectors. Phys Lett A 212:183–187MathSciNetCrossRefMATH

Cirel'son BS (1980) Quantum generalizations of Bell's inequality. Lett Math Phys 4:93–100MathSciNetCrossRef

Clauser JF, Horne MA, Shimony A, Holt RA (1969) Proposed experiment to test local hidden-variable theories. Phys Rev Lett 23:880–884CrossRef

Cleve R, Buhrman H (1997) Substituting quantum entanglement for communication. Phys Rev A 56:1201–1204CrossRef

Cleve R, Høyer P, Toner B, Watrous J (2004) Consequences and limits of nonlocal strategies. Proceedings of the 19th IEEE annual conference on computational complexity, Amherst, MA, pp 236–249

Dakic B, Brukner C (2009) Quantum theory and beyond: is entanglement special? arXiv:0911.0695

DiVincenzo DP, Peres A (1997) Quantum code words contradict local realism. Phys Rev A 55: 4089–4092CrossRef

Einstein A, Podolsky B, Rosen N (1935) Can quantum-mechanical description of physical reality be considered complete? Phys Rev 47:777–780CrossRefMATH

Ekert A (1991) Quantum cryptography based on Bell's theorem. Phys Rev Lett 67:661–663MathSciNetCrossRefMATH

Fannes M, Nachtergaele B, Werner RF (1992) Finitely correlated states on quantum spin chains. Commun Math Phys 144:443–490MathSciNetCrossRefMATH

Galvão EF (2001) Feasible quantum communication complexity protocol. Phys Rev A 65:012318CrossRef

Galvão EF (2002) Foundations of quantum theory and quantum information applications. Ph.D. thesis, University of Oxford, Oxford. Available via arXiv:quant-ph/0212124

Gill RD, Weihs G, Zeilinger A, Żukowski M (2002) No time loophole in Bell's theorem: the Hess-Philipp model is nonlocal. Proc Nat Acad Sci USA 99:14632–14635CrossRefMATH

Gill RD, Weihs G, Zeilinger A, Żukowski M (2003) Comment on “Exclusion of time in the theorem of Bell” by Hess K and Philipp W. Europhy Lett 61:282–283CrossRef

Greenberger DM, Horne MA, Zeilinger A (1989) Going beyond Bell's theorem. In: Kafatos M (ed) Bell's theorem, quantum theory, and conceptions of the universe. Kluwer, Dordrecht, pp 73–76

Greenberger DM, Horne MA, Shimony A, Zeilinger A (1990) Bell's theorem without inequalities. Am J Phys 58:1131–1143MathSciNetCrossRef

Gröblacher S, Paterek T, Kaltenbaek R et al. (2007) An experimental test of non-local realism. Nature 446:871–875CrossRef

Gross D, Flammia S, Eisert J (2009) Most quantum states are too entangled to be useful as computational resources. Phys Rev Lett 102:190501

Holevo AS (1973) Bounds for the quantity of information transmitted by a quantum communication channel. Problemy Peredachi Informatsii 9:3–11. English translation in Probl Inf Transm 9:177–183MathSciNetMATH

Horodecki R, Horodecki P, Horodecki M (1995) Violating Bell inequality by mixed spin-\({1 \over 2}\) states: necessary and sufficient condition. Phys Lett A 200:340–344MathSciNetCrossRefMATH

Hyllus P, Gühne O, Bruß D, Lewenstein M (2005) Relations between entanglement witnesses and Bell inequalities. Phys Rev A 72:012321MathSciNetCrossRef

Kochen S, Specker E (1968) The problem of hidden variables in quantum mechanics. J Math Mech 17:59–87MathSciNet

Kofler J, Paterek T, Brukner C (2006) Experimenter's freedom in Bell's theorem and quantum cryptography, Phys Rev A 73:022104

Kushilevitz E, Nisan N (1997) Communication complexity. Cambridge University Press, CambridgeMATH

Leggett AJ (2002) Testing the limits of quantum mechanics: motivation, state of play, prospects. J Phys Condensed Matter 14:15 R415CrossRef

Leggett AJ (2003) Nonlocal hidden-variable theories and quantum mechanics: an incompatibility theorem. Found Phys 33:1469–1493MathSciNetCrossRef

Leggett AJ, Garg A (1985) Quantum mechanics versus macroscopic realism: is the flux there when nobody looks? Phys Rev Lett 54:857–860MathSciNetCrossRef

Marcovitch S, Reznik B (2008) Implications of communication complexity in multipartite systems. Phys Rev A 77:032120CrossRef

Markov IL, Shi Y (2008) Simulating quantum computation by contracting tensor networks. SIAM J Comput 38(3):963–981MathSciNetCrossRefMATH

Mermin ND (1993) Hidden variables and the two theorems of John Bell. Rev Mod Phys 65:803–815MathSciNetCrossRef

Morikoshi F (2006) Information-theoretic temporal Bell inequality and quantum computation. Phys Rev A 73:052308CrossRef

Nagata K (2005) Kochen-Specker theorem as a precondition for secure quantum key distribution. Phys Rev A 72:012325MathSciNetCrossRef

Pawłowski M, Paterek T, Kaszlikowski D, Scarani V, Winter A and Zukowski M (2009) Information causality as a physical principle. Nature (London) 461(7267):1101–1104

Paz JP, Mahler G (1993) Proposed test for temporal Bell inequalities. Phys Rev Lett 71:3235–3239CrossRef

Peres A (1991) Two simple proofs of the Kochen-Specker theorem. J Phys A 24:L175–L178MathSciNetCrossRef

Peres A (1994) Quantum theory: concepts and methods. Kluwer, Boston

Popescu S, Rohrlich D (1994) Quantum nonlocality as an axiom. Found Phys 24:379–385MathSciNetCrossRef

Rowe MA, Kielpinski D, Meyer V, Sackett CA, Itano WM, Monroe C, Wineland DJ (2001) Experimental violation of a Bell's inequality with efficient detection. Nature 409:791–794CrossRef

Scarani V, Gisin N (2001) Quantum communication between N partners and Bell's inequalities. Phys Rev Lett 87:117901CrossRef

Scarani V, Acín A, Schenck E, Aspelmeyer M (2005) Nonlocality of cluster states of qubits. Phys Rev A 71:042325MathSciNetCrossRef

Schmid C, Kiesel N, Laskowski W et al. (2008) Discriminating multipartite entangled states. Phys Rev Lett 100:200407MathSciNetCrossRef

Scheidl T, Ursin R, Kofler J et al. (2008) Violation of local realism with freedom of choice. Available via http://arxiv.org/abs/0811.3129

Schrödinger E (1935) Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaften 23:807–812; 823–828; 844–849. Translation published in Proc Am Philos Soc 124:323–338 and in Wheeler JA, Zurek WH (eds) (1983) Quantum theory and measurement. Princeton University Press, Princeton, NJ, pp 152–167CrossRef

Shafiee A, Golshani M (2003) Single-particle Bell-type inequality. Annales de la Fondation de Broglie 28:105–118MathSciNet

Simon C, Brukner Č, Zeilinger A (2001) Hidden-variable theorems for real experiments. Phys Rev Lett 86:4427–4430CrossRef

Spekkens RW, Buzacott DH, Keehn AJ et al. (2009) Preparation contextuality powers parity-oblivious multiplexing. Phys Rev Lett 102:010401CrossRef

Svozil K (2004) Quantum mechanics is noncontextual. Available via arXiv:quant-ph/0401112

Tamir B (2007) Communication complexity protocols for qutrits. Phys Rev A 75:032344MathSciNetCrossRef

Tóth G, Gühne O, Briegel HJ (2006) Two-setting Bell inequalities for graph states. Phys Rev A 73:022303MathSciNetCrossRef

Trojek P, Schmid C, Bourennane M et al. (2005) Experimental quantum communication complexity. Phys Rev A 72:050305CrossRef

van Dam W (2000) Implausible consequences of superstrong nonlocality. Chapter 9 in van Dam W, Nonlocality & communication complexity. Ph.D. Thesis, University of Oxford, Department of Physics. Available via arXiv:quant-ph/0501159

Van den Nest M, Miyake A, Dür W, Briegel HJ (2006) Universal resources for measurement-based quantum computation. Phys Rev Lett 97:150504CrossRef

Vidal G (2007) Classical simulation of infinite-size quantum lattice systems in one spatial dimension. Phys Rev Lett 98:070201MathSciNetCrossRef

Weihs G, Jennewein T, Simon C, Weinfurter H, Zeilinger A (1998) Violation of Bell's inequality under strict Einstein locality conditions. Phys Rev Lett 81:5039–5043MathSciNetCrossRefMATH

Weinfurter H, Żukowski M (2001) Four-photon entanglement from down-conversion. Phys Rev A 64:010102CrossRef

Werner RF, Wolf MM (2001) All multipartite Bell correlation inequalities for two dichotomic observables per site. Phys Rev A 64:032112CrossRef

Wiesner S (1983) Conjugate coding, ACM Sigact News 15:78–88CrossRef

von Weizsäcker CF (1985) Aufbau der Physik. Hanser, Munich

Yao AC (1979) Some complexity questions related to distributed computing. Proceedings of the 11th Annual ACM Symposium on Theory of Computing, pp 209–213

Zeilinger A (1999) A foundational principle for quantum mechanics. Found Phys 29:631–643MathSciNetCrossRef

Żukowski M, Brukner Č (2002) Bell's theorem for general N-qubit states. Phys Rev Lett 88:210401MathSciNetCrossRef

Titel: Bell's Inequalities — Foundations and Quantum Communication
verfasst von: Časlav Brukner
Marek Żukowski
Verlag: Springer Berlin Heidelberg
Buch: Handbook of Natural Computing
Print ISBN: 978-3-540-92909-3

Electronic ISBN: 978-3-540-92910-9

Copyright-Jahr: 2012
DOI: https://doi.org/10.1007/978-3-540-92910-9_42

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