Tight steering inequalities from generalized entropic uncertainty relations

TL;DR

This paper develops tight steering inequalities derived from entropic uncertainty relations, enabling efficient and optimal detection of quantum steering in various measurement scenarios using entropy-based criteria.

Contribution

It introduces a general method to construct steering inequalities from entropic uncertainty relations that satisfy specific properties, including Rényi entropies, and demonstrates their tightness and optimality.

Findings

Steering inequalities based on Rényi entropies are tight in many scenarios.

The inequalities precisely recover noise thresholds for steerability with noisy measurements.

Entropy-based criteria can optimally witness quantum steering despite coarse-graining.

Abstract

We establish a general connection between entropic uncertainty relations, Einstein-Podolsky-Rosen steering, and joint measurability. Specifically, we construct steering inequalities from any entropic uncertainty relation, given that the latter satisfies two natural properties. We obtain steering inequalities based on R\'enyi entropies. These turn out to be tight in many scenarios, using max- and min-entropy. Considering steering tests with two noisy measurements, our inequalities exactly recover the noise threshold for steerability. This is the case for any pair of qubit 2-outcome measurements, as well as for pairs of mutually unbiased bases in any dimension. This shows that easy-to-evaluate quantities such as entropy can optimally witness steering, despite the fact that they are coarse-grained representations of the underlying statistics.