arXiv:quant-ph/0312006v1 30 Nov 2003 NOISE AND DISTURBANCE IN QUANTUM MEASUREMENT PAUL BUSCH, TEIKO HEINONEN, AND PEKKA LAHTI Abstract. The operational meaning of some measures of noise and disturbance in measurements is analyzed and their limitations are pointed out. The cases of minimal noise and least disturbance are characterized. 1. Introduction No physical measurement is absolutely accurate. It seems inevitable that there will always be a residual degree of uncertainty as to how close the outcome is to what should have been expected. Likewise, a measurement, being an interaction of the apparatus with the measured system, must always be expected to effect some change, or disturbance, of the measured system. In classical physics it seems possible to achieve arbitrary levels of accuracy and to make the disturbance as small as one wishes. These options appear to be ruled out in quantum physics, due to the fact that there are pairs of physical quantities which cannot be measured together. Such quantities are represented by mutually noncommuting operators or operator measures. In his fundamental work of 1927 on the interpretation of quantum mechanics, W. Heisenberg sketched two versions of what became known as the uncertainty principle and which can be vaguely summarized as follows: (UP1) A measurement, with inaccuracy ǫ(A), of a quantity A that does not commute with a quantity B will disturb the value of B by an amount η(B) such that an appropriate pay-off relation holds between ǫ(A) and η(B). (UP2) A joint measurement of two noncommuting quantities A, B must be imprecise, with the inaccuracies ǫ(A), ǫ(B) satisfying an uncertainty relation. Heisenberg focussed on pairs of canonically conjugate observables and he gave model experiments to demonstrate that relations of the form ǫ(A)η(B) ∼ h and ǫ(A)ǫ(B) ∼ h had to hold in the cases (UP1) and (UP2), respectively. 1