Optimal quantum-state tomography with known parameters

TL;DR

This paper explores optimal quantum-state tomography when some parameters of the quantum state are known, focusing on a 3D optimization problem related to conditional SIC-POVMs, and demonstrates the complexity of designing such measurements.

Contribution

It introduces a numerical approach to optimize quantum measurements with prior information, specifically for conditional SIC-POVMs in three dimensions, advancing the understanding of measurement design with known parameters.

Findings

Solved a 3D optimization problem for conditional SIC-POVMs.

Demonstrated the complexity of measurement optimization with prior information.

Presented numerical methods applicable to various special cases.

Abstract

It is a well-known fact that the optimal POVM for quantum state tomography is the symmetric, informationally complete, positive operator valued measure (SIC-POVM). We investigate the same problem only in the case when there are some a priori information about the state, specifically when some parameters are known. In this paper we mainly focus on solving a 3-dimensional optimization problem, which gives us a non-trivial example for the so-called conditional SIC- POVMs, a straightforward generalization of the concept of SIC-POVMs. We also present other special cases to show further applications of the proposed numerical methods and to illustrate the complexity of this topic.