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

TL;DR

This paper proves the consistency of the quantum-like representation algorithm (QLRA) for hyperbolic interference, ensuring it produces stable probabilistic representations in hyperbolic Hilbert spaces, extending previous results on trigonometric interference.

Contribution

It completes the proof of QLRA's consistency for hyperbolic interference and discusses hyper trigonometric interference, ensuring the algorithm's reliability for complex probabilistic data.

Findings

QLRA is consistent for hyperbolic interference.

Hyper trigonometric interference is addressed.

Representations are unitarily equivalent under natural assumptions.

Abstract

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).