A transform of complementary aspects with applications to entropic uncertainty relations

TL;DR

This paper constructs symmetric sets of mutually unbiased bases in quantum systems, generalizes Fourier transforms, and derives tight entropic uncertainty bounds, advancing understanding in quantum cryptography and phase space symmetries.

Contribution

It introduces a new class of symmetric MUBs in dimension 2^n, generalizes the Fourier transform to multiple bases, and establishes tight entropic uncertainty bounds using symmetry.

Findings

Constructed symmetric MUBs with Clifford algebra symmetry

Derived a tight bound for four MUBs in dimension 4

Linked extrema of Wigner functions to entropic uncertainty relations

Abstract

Even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols they remain ill understood. Here, we construct special sets of up to 2n+1 mutually unbiased bases (MUBs) in dimension d=2^n which have particularly beautiful symmetry properties derived from the Clifford algebra. More precisely, we show that there exists a unitary transformation that cyclically permutes such bases. This unitary can be understood as a generalization of the Fourier transform, which exchanges two MUBs, to multiple complementary aspects. We proceed to prove a lower bound for min-entropic entropic uncertainty relations for any set of MUBs, and show that symmetry plays a central role in obtaining tight bounds. For example, we obtain for the first time a tight bound for four MUBs in dimension d=4, which is attained by an eigenstate of our complementarity transform. Finally, we discuss the relation to other symmetries obtained by transformations in discrete phase space, and note that the extrema of discrete Wigner functions are directly related to min-entropic uncertainty relations for MUBs.