Lüders theorem for unsharp quantum measurements

Paul Busch

Ayan Barui, Mayukha Pal, Prasanta Kumar Panigrahi

Paul Busch, J Singh

A theorem of Lüders states that an ideal measurement of a sharp discrete observable does not alter the statistics of another sharp observable if, and only if, the two observables commute. It will be shown that this statement extends to certain pairs of unsharp observables. Implications for local relativistic quantum theory will be discussed.

Gianpiero Cattaneo, Federico Laudisa

Paul Busch

The conflict between relativistic causality and localizability is analysed in the light of the existence of unsharp localization observables. A theorem due to Schlieder is generalized, showing that the assumption of local commutativity implies the localization observable in question to be unsharp in a strong sense. Furthermore, a recent generalization of a theorem of Lüders is applied to demonstrate that local commutativity is a necessary consequence of Einstein causality even in the case of unsharp observables if they admit local measurements. These findings raise the question of whether localization observables can be measured by means of local operations.

Paul Busch, PEKKA J. LAHTI

Abstract The theory of unsharp quantum measurements is reviewed as a generalization of the von Neumann‐Lüders measurement theory and applied to measurements of continuous quantities. A generalized notion of repeatability is proposed which is applicable even to intrinsically unsharp continuous observables. As an illustration a precise formulation of a quantum non‐demolition (QND) measurement performed on a harmonic oscillator is given.