arXiv subjects

Peter Nyman

Publications and source records attributed to Peter Nyman.

6 recordsLinked to original sources

arXiv2011

Quantum-like representation algorithm for trichotomous observables

We study the problem of representation of statistical data (of any origin) by a complex probability amplitude. This paper is devoted to representation of data collected from measurements of two trichotomous observables. The complexity of the problem eventually increased comparing to with the case of dichotomous observables. We see that only special statistical data (satisfying s number of nonlinear constraints) have the quantum--like representation.

quant-ph↗

arXiv2011

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

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.

quant-ph↗

arXiv2010

On the consistency of the quantum-like representation algorithm for hyperbolic interference

Recently quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov [20]--[28] to solve the so-called "inverse Born's rule problem": to construct a representation of probabilistic data by a complex or more general (in particular, hyperbolic) probability amplitude which matches Born's rule or its generalizations. The outcome from QLRA is coupled to the formula of total probability with an additional term corresponding to trigonometric, hyperbolic or hyper-trigonometric interference. The consistency of QLRA for probabilistic data corresponding to trigonometric interference was recently proved [29]. We now complete the proof of the consistency of QLRA to cover hyperbolic interference as well. We will also discuss hyper trigonometric interference. The problem of consistency of QLRA arises, because formally the output of QLRA depends on the order of conditioning. For two observables (e.g., physical or biological) 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 over the hyperbolic algebra). We prove that under "natural assumptions" these two representations are unitary equivalent (in the sense of hyperbolic Hilbert space).

quant-ph↗

arXiv2009

On consistency of the quantum-like representation algorithm

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.

math-ph↗

arXiv2008

Simulation of Quantum Error Correcting Code

This study considers implementations of error correction in a simulation language on a classical computer. Error correction will be necessarily in quantum computing and quantum information. We will give some examples of the implementations of some error correction codes. These implementations will be made in a more general quantum simulation language on a classical computer in the language Mathematica. The intention of this research is to develop a programming language that is able to make simulations of all quantum algorithms and error corrections in the same framework. The program code implemented on a classical computer will provide a connection between the mathematical formulation of quantum mechanics and computational methods. This gives us a clear uncomplicated language for the implementations of algorithms.

quant-ph↗

arXiv2007

Simulation of Quantum Algorithms with a Symbolic Programming Language

This study examines the simulation of quantum algorithms on a classical computer. The program code implemented on a classical computer will be a straight connection between the mathematical formulation of quantum mechanics and computational methods. The computational language will include formulations such as quantum state, superposition and quantum operator.

quant-ph↗