The best approximation of an objective state with a given set of quantum states

TL;DR

This paper derives a closed-form solution for the minimal l_2 norm distance between a target quantum state and convex combinations of a given set of states, with implications for quantum resource theory.

Contribution

It provides the first explicit formula for the minimal convex approximation distance and the minimal number of states needed for optimal approximation.

Findings

Closed-form solution for minimal convex approximation distance.

Numerical verification with randomly generated quantum states.

Determination of the minimal number of states for optimal approximation.

Abstract

Approximating a quantum state by the convex mixing of some given states has strong experimental significance and provides potential applications in quantum resource theory. Here we find a closed form of the minimal distance in the sense of l_2 norm between a given d-dimensional objective quantum state and the state convexly mixed by those restricted in any given (mixed-) state set. In particular, we present the minimal number of the states in the given set to achieve the optimal distance. The validity of our closed solution is further verified numerically by several randomly generated quantum states.