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Representation of probabilistic data by complex probability amplitudes: the case of triple-valued observables.
Linnéuniversitetet, Fakultetsnämnden för naturvetenskap och teknik, Institutionen för datavetenskap, fysik och matematik, DFM.
2010 (Engelska)Ingår i: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 1327, s. 439-449Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

The problem of inter-relation between classical and quantum probabilistic data wasdiscussed in numerous papers (from various points of view), see, e.g., [1, 2, 3, 4, 6, 5, 7,8, 14, 15]. We are interested in the problem of representation of probabilistic data of anyorigin 1 by complex probability amplitude, so to say a “wave function”. This problemwas discussed in very detail in [17]. A general QL-representation algorithm (QLRA)was presented in [17]. This algorithm is based on the formula of total probability withinterference term – a disturbance of the standard formula of total probability. Startingwith experimental probabilistic data, QLRA produces a complex probability amplitudesuch that probability can be reconstructed by using Born’s rule.Although the formal scheme of QLRA works for multi-valued observables of anarbitrary dimension, the description of the class of probabilistic data which can betransfered into QL-amplitudes (the domain of application of QLRA) depends very muchon the dimension. In [19] the simplest case of data generated by dichotomous observableswas studied. In this paper we study trichotomous observables. The complexity of theproblem increases incredibly comparing with the two dimensional case.Finally, we remark that our study is closely related to the triple slit interferenceexperiment and Sorkin’s equality [16]. This experiment provides an important test offoundations of QM.The scheme of presentation is the following one. We start with observables given byQM and derive constraints on phases which are necessary and sufficient for the QLrepresentation.Then we use these constraints to produce complex amplitudes from data(of any origin); some examples, including numerical, are given.

Ort, förlag, år, upplaga, sidor
American Institute of Physics (AIP), 2010. Vol. 1327, s. 439-449
Nationell ämneskategori
Matematik
Forskningsämne
Naturvetenskap, Matematik
Identifikatorer
URN: urn:nbn:se:lnu:diva-13821ISBN: 978-0-7354-0882-1 (tryckt)OAI: oai:DiVA.org:lnu-13821DiVA, id: diva2:435378
Konferens
International Conference Advances in Quantum Theory Location: Vaxjo, SWEDEN Date: JUN 14-17, 2010
Tillgänglig från: 2011-08-18 Skapad: 2011-08-18 Senast uppdaterad: 2018-05-16Bibliografiskt granskad
Ingår i avhandling
1. On relations between classical and quantum theories of information and probability
Öppna denna publikation i ny flik eller fönster >>On relations between classical and quantum theories of information and probability
2011 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

In this thesis we study quantum-like representation and simulation of quantum algorithms by using classical computers.The quantum--like representation algorithm (QLRA) was  introduced by A. Khrennikov (1997) to solve the ``inverse Born's rule problem'', i.e. to construct a representation of probabilistic data-- measured in any context of science-- and represent this data by a complex or more general probability amplitude which matches a generalization of Born's rule.The outcome from QLRA matches the formula of total probability with an additional trigonometric, hyperbolic or hyper-trigonometric interference term and this is in fact a generalization of the familiar formula of interference of probabilities.

We study representation of statistical data (of any origin) by a probability amplitude in a complex algebra and a Clifford algebra (algebra of hyperbolic numbers). The statistical data is collected from measurements of two dichotomous and trichotomous observables respectively. We see that only special statistical data (satisfying a number of nonlinear constraints) have a quantum--like representation.

We also study simulations of quantum computers on classical computers.Although it can not be denied that great progress have been made in quantum technologies, it is clear that there is still a huge gap between the creation of experimental quantum computers and realization of a quantum computer that can be used in applications. Therefore the simulation of quantum computations on classical computers became an important part in the attempt to cover this gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. Of course, it can not be expected that quantum algorithms would help to solve NP problems for polynomial time on classical computers. However, this is not at all the aim of classical simulation.

 The second part of this thesis is devoted to adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulations on classical computers. Concretely we represent Simon's algorithm, Deutsch-Josza algorithm, Shor's algorithm, Grover's algorithm and quantum error-correcting codes in the Mathematica symbolic language. We see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include future algorithms in this framework.

Ort, förlag, år, upplaga, sidor
Växjö, Kalmar: Linnaeus University Press, 2011. s. 161
Serie
Linnaeus University Dissertations ; 60/2011
Nyckelord
Born’s rule, Clifford algebra, Deutsch-Josza algorithm, Grover’s algorithm, Hyperbolic interferences, Inverse Born’s rule problem, Probabilistic data, Quantum computing, Quantum error-correcting, Quantum-like representation algorithm, Shor’s algorithm, Simon’s algorithm, Simulation of quantum algorithms
Nationell ämneskategori
Matematik
Forskningsämne
Naturvetenskap, Matematik
Identifikatorer
urn:nbn:se:lnu:diva-13830 (URN)978-91-86491-98-7 (ISBN)
Disputation
2011-09-22, Weber, Universitetsplatsen 1, Växjö, 14:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2011-08-18 Skapad: 2011-08-18 Senast uppdaterad: 2011-08-18Bibliografiskt granskad

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