Optimal convex approximations of quantum states

TL;DR

This paper addresses the problem of optimally approximating a quantum state by convex combinations of available states, providing a complete solution for qubits using Pauli bases.

Contribution

It introduces a method to find the best convex approximation of quantum states and offers a complete solution for qubits with Pauli basis states.

Findings

Complete solution for qubit state approximation

Optimal weights for convex mixing derived

Improved understanding of quantum state approximation

Abstract

We consider the problem of optimally approximating an unavailable quantum state $\rho $ by the convex mixing of states drawn from a set of available states $\{ \nu_i\}$. The problem is recast to look for the least distinguishable state from $\rho $ among the convex set $\sum_i p_i \nu_i$, and the corresponding optimal weights $\{ p_i \}$ provide the optimal convex mixing. We present the complete solution for the optimal convex approximation of a qubit mixed state when the set of available states comprises the three bases of the Pauli matrices.