Quantum uncertainty relation based on the mean deviation

TL;DR

This paper introduces alternative quantum uncertainty relations based on mean deviation, offering robustness in scenarios where standard deviation approaches fail, and demonstrates their effectiveness in entanglement detection and EPR violation under loss.

Contribution

It presents new mean deviation-based uncertainty relations, both state dependent and independent, expanding the tools for quantum uncertainty analysis beyond standard deviation.

Findings

Mean deviation based uncertainty relations are robust in loss scenarios.

They can detect EPR violations at higher inefficiency thresholds.

They perform comparably to standard deviation methods in entanglement detection.

Abstract

Traditional forms of quantum uncertainty relations are invariably based on the standard deviation. This can be understood in the historical context of simultaneous development of quantum theory and mathematical statistics. Here, we present alternative forms of uncertainty relations, in both state dependent and state independent forms, based on the mean deviation. We illustrate the robustness of this formulation in situations where the standard deviation based uncertainty relation is inapplicable. We apply the mean deviation based uncertainty relation to detect EPR violation in a lossy scenario for a higher inefficiency threshold than that allowed by the standard deviation based approach. We demonstrate that the mean deviation based uncertainty relation can perform equally well as the standard deviation based uncertainty relation as non-linear witness for entanglement detection.