Efficient Quantum Tomography Needs Complementary and Symmetric Measurements

In this study the determinant of the average quadratic error matrix is used as the measure of state estimation efficiency. Minimizing this quantity gives us the optimal measurements in different scenarios. We present applications when von Neumann measurements or a single positive operator-valued measure are used, when there is no known information or a part of the parameters of the state is given. Under some restrictions the optimality is found for n-level systems. The optimal measurements have some complementary relation to each other and to the available data, moreover, symmetric informationally complete systems appear, containing a new, conditional version.