Minimal-disturbance measurement as a specification in von Neumann's quantal theory of measurement

Abstract

In von Neumann's theory an incomplete observableA is measured by measuring any complete observableB whose functionA is. This procedure is narrowed down in this paper by the additional requirement of preservation of the sharp value of any observable compatible withA. The requirement is shown to be equivalent to the unique change of state:ρ → (trρPn)−1PnρPn (Pn is the eigenprojector ofA corresponding to the obtained eigenvaluean, ρ is the statistical operator of the initial state, and by assumption trρPn > 0). This characterises the minimal-disturbance measurement. A necessary and sufficient condition is derived for the selection of the above observableB so that its measurement implies the minimal-disturbance measurement ofA. For arbitraryρ andA, there exists aB satisfying the condition. Hence, this constitutes a reasonable specification within von Neumann's theory, reducing the latter to the physically preferable minimal-disturbance measurement theory.