Complete Optimal Convex Approximations of Qubit States under $B_2$ Distance

TL;DR

This paper derives analytical formulas and optimal decompositions for approximating any qubit state using eigenstates of two Pauli matrices under the $B_2$-distance, exploring tradeoff relations.

Contribution

It provides the first complete analytical solutions for optimal convex approximations of qubit states under the $B_2$-distance measure.

Findings

Analytical formulas for $B_2$-distance and optimal decompositions

Tradeoff relations for sum and squared sum of distances

Numerical validation of analytical results

Abstract

We consider the optimal approximation of arbitrary qubit states with respect to an available states consisting the eigenstates of two of three Pauli matrices, the $B_2$-distance of an arbitrary target state. Both the analytical formulae of the $B_2$-distance and the corresponding complete optimal decompositions are obtained. The tradeoff relations for both the sum and the squared sum of the $B_2$-distances have been analytically and numerically investigated.