Quantumness, generalized 2-desing and symmetric informationally complete POVM

C. A. Fuchs and M. Sasaki defined the quantumness of a set of quantum states in [1], which is related to the fidelity loss in transmission of the quantum states through a classical channel. In [4], Fuchs showed that in d-dimensional Hilbert space, minimum quantumness is 2 d+1 , and this can be achieved by all rays in the space. He left an open problem, asking whether fewer than d states can achieve this bound. Recently, in a different context, A. J. Scott introduced a concept of generalized t-design in [2], which is a natural generalization of spherical t-design. In this paper, we show that the lower bound on the quantumness can be achieved if and only if the states form a generalized 2-design. As a corollary, we show that this bound can be only achieved if the number of states are larger or equal to d, answering the open problem. Furthermore, we also show that the minimal set of such ensemble is Symmetric Informationally Complete POVM(SICPOVM). This leads to an equivalence relation between SIC-POVM and minimal set of ensemble achieving minimal quantumness.