Robertson-Schrödinger-type formulation of Ozawa's noise-disturbance uncertainty principle

In this work we derive a matrix formulation of a noise-disturbance uncertainty relation, which is akin to the Robertson-Schrödinger uncertainty principle. Our inequality is stronger than Ozawa's uncertainty principle and takes noise-disturbance correlations into account. Moreover, we show that for certain types of measurement interactions it is covariant with respect to linear symplectic transformations of the noise and disturbance operators. Finally, we also study the tightness of our matrix inequality.