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

TL;DR

This paper explores hyperbolic interferences in quantum-like representations of probabilistic data, especially for triple-valued observables, extending the framework to Clifford algebras and identifying conditions for data to admit such representations.

Contribution

It introduces a class of statistical data satisfying nonlinear constraints that allow quantum-like representations with hyperbolic interferences for triple-valued observables.

Findings

Identifies conditions for data to have quantum-like representations

Extends the framework to Clifford algebra for hyperbolic numbers

Demonstrates representation for data from two trichotomous observables

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.