Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement

The Heisenberg uncertainty principle states that the product of the noise in a position measurement and the momentum disturbance caused by that measurement should be no less than the limit set by Planck’s constant $ħ/2\mathrm{}$ as demonstrated by Heisenberg’s thought experiment using a γ-ray microscope. Here it is shown that this common assumption is not universally true: a universally valid trade-off relation between the noise and the disturbance has an additional correlation term, which is redundant when the intervention brought by the measurement is independent of the measured object, but which allows the noise-disturbance product much below Planck’s constant when the intervention is dependent. A model of measuring interaction with dependent intervention shows that Heisenberg’s lower bound for the noise-disturbance product is violated even by a nearly nondisturbing precise position measurement. An experimental implementation is also proposed to realize the above model in the context of optical quadrature measurement with currently available linear optical devices.