Entropic uncertainty relations for multiple measurements assigned with biased weights

TL;DR

This paper develops weighted Rényi entropic uncertainty relations for quantum measurements with nonuniform probabilities, providing stronger bounds and practical optimization methods that enhance quantum information tasks without additional quantum resources.

Contribution

It introduces state-dependent weighted EURs for multiple measurements, extending traditional EURs and demonstrating their advantages through numerical verification.

Findings

Weighted EURs are generally stronger than previous bounds.

Optimization of measurement weights improves quantum task performance.

The proposed method is computationally efficient and resource-free.

Abstract

The entropic way of formulating Heisenberg's uncertainty principle not only plays a fundamental role in applications of quantum information theory but also is essential for manifesting genuine nonclassical features of quantum systems. In this paper we investigate R\'{e}nyi entropic uncertainty relations (EURs) in the scenario where measurements on individual copies of a quantum system are selected with nonuniform probabilities. In contrast with EURs that characterize an observer's overall lack of information about outcomes with respect to a collection of measurements, we establish state-dependent lower bounds on the weighted sum of entropies over multiple measurements. Conventional EURs thus correspond to the special cases when all weights are equal, and in such cases, we show our results are generally stronger than previous ones. Moreover, taking the entropic steering criterion as an example, we numerically verify that our EURs could be advantageous in practical quantum tasks by optimizing the weights assigned to different measurements. Importantly, this optimization does not require quantum resources and is efficiently computable on classical computers.