Optimal convex approximations of quantum states based on fidelity

TL;DR

This paper develops a method for optimally approximating quantum states using convex combinations based on fidelity, providing exact solutions for qubits and analyzing geometric properties of target states.

Contribution

It introduces a fidelity-based approach for optimal convex approximation of quantum states and offers complete solutions for arbitrary qubit states.

Findings

Fidelity-based optimal states are closer to target than trace norm-based states in many cases.

Complete exact solutions are provided for arbitrary qubit states.

The geometric properties of target states can be fully characterized by a set of three available states.

Abstract

We investigate the problem of optimally approximating a desired state by the convex mixing of a set of available states. The problem is recasted as finding the optimal state with the minimum distance from target state in a convex set of usable states. Based on the fidelity, we define the optimal convex approximation of an expected state and present the complete exact solutions with respect to an arbitrary qubit state. We find that the optimal state based on fidelity is closer to the target state than the optimal state based on trace norm in many ranges. Finally, we analyze the geometrical properties of the target states which can be completely represented by a set of practicable states. Using the feature of convex combination, we express this class of target states in terms of three available states.