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

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.

Optimal convex approximation of qubit states and geometry of completely represented states

“…This study depends not only on the set of available states but also on the distance measures between quantum states. There were some work based on the trace norm [12][13][14][15]17] and fidelity [16].…”

“…In this letter, we investigate the optimal convex approximation of desired qubit states with respect to the convex mixing of a set of available states based on the fidelity [16,29,30]. In the next section, we provide the complete exact solution for optimal convex approximation of any qubit state in regard to the set B 3 which represents the set of eigenvectors of three Pauli matrices.…”

“…, n}, where |ψ i are available pure states. Based on the fidelity, we introduce the following definition [16].…”