Accessible Information and Informational Power of Quantum 2-designs

The accessible information and the informational power quantify the amount of information extractable from a quantum ensemble and by a quantum measurement, respectively. So-called spherical quantum 2-designs constitute a class of ensembles and measurements relevant in testing entropic uncertainty relations, quantum cryptography, and quantum tomography. We provide lower bounds on the entropy of 2-design ensembles and measurements, from which upper bounds on their accessible information and informational power follow, as a function of the dimension only. We show that the statistics generated by 2-designs, although optimal for the abovementioned protocols, never contains more than one bit of information. Finally, we specialize our results to the relevant cases of symmetric informationally complete (SIC) sets and maximal sets of mutually unbiased bases (MUBs), and we generalize them to the arbitrary-rank case.