Entropic Bounds For Unitary Testers and Mutually Unbiased Unitary Bases

TL;DR

This paper introduces entropic bounds for unitary testers, explores their relation to mutually unbiased unitary bases, and discusses implications for quantum cryptography, highlighting the role of maximal bounds and unbiased bases.

Contribution

It defines entropic bounds for unitary testers, characterizes sets saturating these bounds, and links mutually unbiased unitary bases to maximal entropic bounds in quantum information.

Findings

Sets saturating zero bounds have identical statistics.

Maximal bounds are achieved by mutually unbiased unitary bases.

These bounds have applications in quantum cryptographic protocols.

Abstract

We define the entropic bounds, i.e minimal uncertainty for pairs of unitary testers in distinguishing between unitary transformations not unlike the well known entropic bounds for observables. We show that in the case of specific sets of testers which pairwise saturate the trivial zero bound, the testers are all equivalent in the sense their statistics are the same. On the other hand, when maximal bounds are saturated by such sets of testers, the unitary operators would form unitary bases which are mutually unbiased. This resembles very much the role of mutually unbiased bases in maximizing the entropic bounds for observables. We show how such a bound can be useful in certain quantum cryptographic protocols.