Triangle geometry for qutrit states in the probability representation

TL;DR

This paper introduces a geometric framework for representing qutrit states using triangle geometry and probability distributions, establishing a formal link between quantum and classical statistics for three-level systems.

Contribution

It develops a novel triangle geometry approach for qutrit states, mapping Bloch sphere geometry to triangle configurations and expressing quantum channels as linear probability transforms.

Findings

Qutrit density matrix elements are expressed via probabilities of artificial qubit states.

Quantum statistics of qutrits is shown to be equivalent to classical systems with constraints.

Triangle geometry and inequalities for areas of Malevich's squares are derived for qutrit states.

Abstract

We express the matrix elements of the density matrix of the qutrit state in terms of probabilities associated with artificial qubit states. We show that the quantum statistics of qubit states and observables is formally equivalent to the statistics of classical systems with three random vector variables and three classical probability distributions obeying special constrains found in this study. The Bloch spheres geometry of qubit states is mapped onto triangle geometry of qubits. We investigate the triada of Malevich's squares describing the qubit states in quantum suprematism picture and the inequalities for the areas of the squares for qutrit (spin-1 system). We expressed quantum channels for qutrit states in terms of a linear transform of the probabilities determining the qutrit-state density matrix.