Efficient quantum tomography needs complementary and symmetric measurements

TL;DR

This paper explores how using complementary and symmetric measurements can optimize quantum state estimation, providing practical methods for designing measurement setups in quantum tomography.

Contribution

It introduces a new approach to optimize quantum tomography using the determinant of the average quadratic error matrix, highlighting the role of complementary and symmetric measurements.

Findings

Optimal measurements are complementary or symmetric.

Symmetric informationally complete systems are identified as optimal.

The approach applies to single qubits and n-level systems.

Abstract

In this study the determinant of the average quadratic error matrix is used as the measure of state estimation efficiency. This quantity is easily computable in some cases, so it gives us a reasonable tool to find optimal measurement setup for different quantum tomography problems. We present numerous applications for a single qubit when von Neumann measurements or a single POVM is used and a part of the parameters of the state is given. Under some restriction the optimality is found for $n$-level system as well. The optimal measurements have some complementary relation to each other or to the known datas, moreover, symmetric informationally complete systems appear, the conditional version seems to be new.