Quantum uncertainty relation based on the mean deviation

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.

Figure 1

Graphical depiction of the efficacy of MD-based uncertainty relations. (a) Physical potential $V\left(x\right)$ as a function of position $x$ corresponding to the $F$-distribution probability density. Here we have set $V_0=0$ and $\hslash =m=1$ and $d_1=1$. (b) Probability distribution function for the $F$ distribution for various $p$ values of $d_2$ ($d_1=1$). The blue striped zone, with an example depicted by the blue line corresponding to $d_2=5$, is where both SD-based uncertainty relations and MD-based uncertainty relations are applicable. The white zone, with an example depicted by the green line corresponding to $d_2=3$, is where the MD-based uncertainty relations apply but SD-based ones do not. The red squared zone, with an example furnished by the red line corresponding to $d_2=1$, is where both the MD- and SD-based uncertainty relations fail to apply. The lines corresponding to $d_2=2$ (dot-dashed) and $d_2=4$ (dashed) set the boundaries between these zones. We set $d_1=1$ throughout.

Figure 2

Schematic diagram for the lossy detector scenario. The two-qubit initial Werner state is written in terms of a $9\times 9$ density matrix ${\rho }_F$ taking into account particle loss before measurement.

Figure 3

Illustration of the power of MD-based uncertainty relations in the lossy EPR violation scenario. The SD-based uncertainty relations cannot detect EPR violation if the detector efficiency $\eta$ plotted along the $y$ axis (with respect to the Werner noise parameter $p$) is below the dashed blue curve. However, up to the limit of the solid maroon curve, i.e., in the maroon-striped zone, the MD-based uncertainty relation can still detect such EPR violation.