Accessible Information and Informational Power of Quantum 2-designs

TL;DR

This paper establishes bounds on the information extractable from quantum 2-designs, showing they never carry more than one bit of information, with implications for quantum cryptography and tomography.

Contribution

It provides entropy bounds for quantum 2-designs and their measurements, extending results to SIC sets and MUBs, and generalizing to arbitrary-rank cases.

Findings

2-designs generate at most one bit of information

Lower bounds on entropy lead to bounds on accessible information

Results apply to SIC sets, MUBs, and arbitrary-rank ensembles

Abstract

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.