Quantum uncertainty relation based on the mean deviation

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 for a general set of deviation measures, with special emphasis 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 Einstein-Podolsky-Rosen 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 the nonlinear witness for entanglement detection.