On consistency of the quantum-like representation algorithm

TL;DR

This paper proves that the quantum-like representation algorithm (QLRA) produces consistent, unitarily equivalent representations of probabilistic data, resolving an important aspect of the inverse Born's rule problem in quantum-like modeling.

Contribution

It demonstrates the unitary equivalence of different representations generated by QLRA, confirming its internal consistency under natural assumptions.

Findings

QLRA representations are unitarily equivalent

Consistency of QLRA is established

Supports the validity of quantum-like modeling methods

Abstract

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.