The best approximation of a given qubit state with the limited pure-state set

TL;DR

This paper investigates the optimal approximation of a target qubit state using a limited set of pure states, providing analytical solutions and methods to reduce the problem to manageable cases.

Contribution

It introduces an analytical approach for optimal qubit state approximation with up to three pure states and reduces larger state sets to these cases.

Findings

Analytical optimal distance based on fidelity derived.

Preparation with more than four states can be reduced to three or fewer states.

Results validated through comparison of analytical and numerical methods.

Abstract

The preparation of quantum states lies at the foundation in the quantum information processing. The convex mixing of some existing quantum states is one of the effective candidate. In this paper, we mainly study how a target quantum state can be optimally prepared by not more than three given pure states. The analytic optimal distance based on the fidelity is found. We also show that the preparation with more than four states can be essentially converted to the case with not more than four states, which can be similarly solved as the case with three states. The validity is illustrated by the comparison of our analytical and numerical results.