On simultaneous measurements of incompatible observables

Ittoop Vergheese Puthoor

Willem M. de Muynck, J. M. V. A. Koelman

Eduard Prugovečki

The empirical meaning of the simultaneity of measurement of several observables is carefully analyzed. General criteria sufficient to guarantee the internal consistency of a framework including simultaneous measurements of incompatible observables are put forward. The case is illustrated with the example of simultaneous measurements of position and momenta of particles in quantum mechanics. It is pointed out that, in general, the outcome of measurements of incompatible observables cannot be described statistically by means of the probability measures of mathematical statistics, which have primarily been introduced only with compatible observables in mind. The concept of complex probability measure is put forward. A generalization of the current interpretation of quantum mechanics is proposed that would also include the case of incompatible observables, by taking advantage of the concept of complex probability measure.

Jeramy Gordon, W. H. Louisell

Teiko Heinosaari, Maria Anastasia Jivulescu, Daniel Reitzner +1 more

We study the problem of performing orthogonal qubit measurements simultaneously. Since these measurements are incompatible, one has to accept additional imprecision. An optimal joint measurement is the one with the least possible imprecision. All earlier considerations of this problem have concerned only joint measurability of observables, while in this work we also take into account conditional state transformations (i.e., instruments). We characterize the optimal joint instrument for two orthogonal von Neumann instruments as being the L\"uders instrument of the optimal joint observable.

Shiro Ishikawa