Asymptotic entropic uncertainty relations

TL;DR

This paper establishes strong, optimal lower bounds on the average Shannon entropy for measurements in random bases in high-dimensional quantum systems, confirming conjectures about asymptotic behavior of entropic uncertainty constants.

Contribution

It introduces new bounds on entropic uncertainty relations for random bases, proving their optimality and confirming asymptotic conjectures in high dimensions.

Findings

Stronger entropy bounds than Maassen-Uffink's relation.

Bounds are optimal up to additive constants.

Confirmed the Wehner-Winter conjecture asymptotically.

Abstract

We analyze entropic uncertainty relations for two orthogonal measurements on a $N$-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix $U$ relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than these obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to establish the conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of an independent interest.