Construction of general symmetric-informationally-complete–positive-operator-valued measures by using a complete orthogonal basis

A general symmetric-informationally-complete (GSIC)–positive-operator-valued measure (POVM) is known to provide an optimal quantum state tomography among minimal IC POVMs with a fixed average purity. In this paper we provide a general construction of a GSIC POVM by means of a complete orthogonal basis (COB), also interpreted as a normal quasiprobability representation. A spectral property of a COB is shown to play a key role in the construction of SIC POVMs and also for the bound of the mean-square error of the state tomography. In particular, a necessary and sufficient condition to construct a SIC POVM for any $d$ is constructively given by the power of traces of a COB. We give three simple constructions of COBs from which one can systematically obtain GSIC POVMs.

Figure 1

Value ${\lambda }^{*}$ of COBs made by Construction t9 and the optimal value $\frac{1}{\sqrt{d+1}}$.