On the quantumness of a Hilbert space

We derive an exact expression for the quantumness of a Hilbert space (defined in C. A. Fuchs and M. Sasaki, Quant. Info. Comp. 3, 377 (2003)), and show that in composite Hilbert spaces the signal states must contain at least some entangled states in order to achieve such a sensitivity. Furthermore, we establish that the accessible fidelity for symmetric informationally complete signal ensembles is equal to the quantumness. Though spelling the most trouble for an eavesdropper because of this, it turns out that the accessible fidelity is nevertheless easy for her to achieve in this case: Any measurement consisting of rank-one POVM elements is an optimal measurement, and the simple pro- cedure of reproducing the projector associated with the measurement outcome is an optimal output strategy.