Uncertainty relation and the measurement error–perturbation relation

The origins and physical consequences of the traditionally used relation between the position measurement error and the momentum perturbation, $\Delta_m^2x\Delta_p^2p\ge\hbar^2/4$ are discussed. It is demonstrated that the corresponding increase in the momentum variance for the aposteriori state occurs only in some special cases. The product of $\Delta_m^2A$ and $\Delta_p^2B$ is shown to essentially differ from the one given by the uncertainty relation if the commutator $[\hat A,\hat B]$ is an operator. The error quantum limits for the joint homodyne measurement of quadrature amplitudes for an optical mode are found. It is shown that similar results can be obtained if the quadratures of a harmonic oscillator are estimated by means of continuous position measurement.