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arXiv:1108.2194

On hyperbolic interferences in the quantum--like representation algorithm for the case of triple-valued observables

Peter Nyman

Published Aug 10, 2011• 2 citations
arXivHTMLPDF

Abstract

The quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov \cite{K1,K2,K3,K4,K5} 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 (A Clifford algebra is introduced for this more general representation) probability amplitude which matches a generalization of Born's rule. The outcome from QLRA will introduce the formula of total probability with an additional term of trigonometric, hyperbolic or hyper-trigonometric interference 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 datas are collected from measurements of two trichotomous observables and the complexity of the problem increased eventually compared to the case of dichotomous observables. We see that only special statistical data (satisfying a number of nonlinear constraints) have a quantum-like representation. In this paper we will present a class of statistical data which satisfy these nonlinear constraints and have a quantum-like representation. This quantum-like representation induces trigonometric-, hyperbolic- and hyper-trigonometric interferences representation.

DOI
10.48550/arxiv.1108.2194

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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.
In this paper we continue to study so called “inverse Born’s rule problem”: to construct representation of probabilistic data of any origin by a complex probability amplitude which matches Born’s rule. The corresponding algorithm – quantum-like representation algorithm (QLRA) was recently proposed by A. Khrennikov [1]–[5]. Formally QLRA depends on the order of conditioning. For two observables a and b, b|a- and a|b conditional probabilities produce two representations, say in Hilbert spaces H b|a and H a|b. In this paper we prove that under natural assumptions these two representations are unitary equivalent. This result proves consistency QLRA. Keywords quantum-like representation algorithm, inverse Born’s rule problem, order of conditioning, unitary equivalence of representations. 1
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