Sequences of lower bounds for entropic uncertainty relations from bistochastic maps

TL;DR

This paper introduces a new method for deriving sequences of lower bounds for entropic uncertainty relations using bistochastic maps, which can be efficiently applied to high-dimensional quantum systems.

Contribution

The authors propose a novel strategy based on bistochastic maps to generate sequences of lower bounds for entropic uncertainty, improving analysis of observable incompatibility.

Findings

Method provides advantage for pure and mixed states.

Sequences are computationally efficient for high-dimensional systems.

Applicable to various examples demonstrating effectiveness.

Abstract

Given two orthornormal bases A and B, the basic form of the entropic uncertainty principle is stated in terms of the sum of the Shannon entropies of the probabilities of measuring A and B onto a given quantum state. State independent lower bounds for this sum encapsulate the degree of incompatibility of the observables diagonal in the A and B bases, and are usually derived by extracting as much information as possible from the unitary operator U connecting the two bases. Here we show a strategy to derive sequences of lower bounds based on alternating sequences of measurements onto A and B. The problem can be mapped into the multiple application of bistochastic processes that can be described by the powers of the unistochastic matrices directly derivable from U. By means of several examples we study the applicability of the method. The results obtained show that the strategy can allow for an advantage both in the pure state and in the mixed state scenario. The sequence of lower bounds is obtained with resources which are polynomial in the dimension of the underlying Hilbert space, and it is thus suitable for studying high dimensional cases.