Pattern Recognition In Non-Kolmogorovian Structures

TL;DR

This paper generalizes pattern recognition to non-Kolmogorovian probabilistic models, including quantum systems, enabling the use of quantum correlations like entanglement for improved recognition tasks.

Contribution

It introduces a rigorous framework for pattern recognition in non-Kolmogorovian models, encompassing classical and quantum cases, and explores the computational complexity involved.

Findings

Framework includes quantum correlations such as entanglement and discord.

Quantum pattern recognition problems relate to key quantum algorithms.

Provides examples illustrating the application of the framework.

Abstract

We present a generalization of the problem of pattern recognition to arbitrary probabilistic models. This version deals with the problem of recognizing an individual pattern among a family of different species or classes of objects which obey probabilistic laws which do not comply with Kolmogorov's axioms. We show that such a scenario accommodates many important examples, and in particular, we provide a rigorous definition of the classical and the quantum pattern recognition problems, respectively. Our framework allows for the introduction of non-trivial correlations (as entanglement or discord) between the different species involved, opening the door to a new way of harnessing these physical resources for solving pattern recognition problems. Finally, we present some examples and discuss the computational complexity of the quantum pattern recognition problem, showing that the most important quantum computation algorithms can be described as non-Kolmogorovian pattern recognition problems.