Triangle geometry of the qubit state in the probability representation expressed in terms of the triada of Malevich's Squares

TL;DR

This paper introduces a geometric triangle representation of qubit states using Malevich's squares, linking state evolution to triangle vertex motion and providing a new perspective on qubit dynamics and uncertainty relations.

Contribution

It presents a novel geometric framework mapping qubit states onto a triangle with vertices related to Malevich's squares, connecting state evolution to triangle vertex motion.

Findings

Triangle vertices correspond to points inside the Bloch sphere.

Uncertainty bounds relate to the area of the triangle and Malevich's squares.

Qubit state evolution can be visualized as vertex motion within the triangle.

Abstract

We map the density matrix of the qubit (spin-1/2) state associated with the Bloch sphere and given in the tomographic probability representation onto vertices of a triangle determining Triada of Malevich's squares. The three triangle vertices are located on three sides of another equilateral triangle with the sides equal to $\sqrt 2$. We demonstrate that the triangle vertices are in one-to-one correspondence with the points inside the Bloch sphere and show that the uncertainty relation for the three probabilities of spin projections +1/2 onto three orthogonal directions has the bound determined by the triangle area introduced. This bound is related to the sum of three Malevich's square areas where the squares have sides coinciding with the sides of the triangle. We express any evolution of the qubit state as the motion of the three vertices of the triangle introduced and interpret the gates of qubit states as the semigroup symmetry of the Triada of Malevich's squares. In view of the dynamical semigroup of the qubit-state evolution, we constructed nonlinear representation of the group U(2).