Unsharp observables and their joint measurement

Nai-Le Liu, Li Li, Sixia Yu +1 more

Sixia Yu, Nai-Le Liu, Li Li +1 more

We derive a single inequality as the sufficient and necessary condition for two unsharp observables of a two-level system to be jointly measurable in a single apparatus and construct explicitly the joint observables of two jointly measurable observables. By introducing a generalized distinguishability as a measure of the unsharpness of an unsharp measurement, we derive a complementarity inequality, which generalizes Englert's duality inequality, from the condition of joint measurement of two orthogonal unsharp observables.

CATTANEO G., Giuseppe Antonio NISTICO'

Rajarshi Pal, Sibasish Ghosh A

We consider a joint measurement of two and three unsharp qubit observables through an Arthur–Kelly-type joint measurement model for qubits. We investigate the effect of the initial state of the detectors on the unsharpness of the measurement as well as the post-measurement state of the system. Particular emphasis is given on a physical understanding of the POVM to PVM transition in the model and entanglement between the system and the detectors. Two approaches for characterizing the unsharpness of the measurement and the resulting measurement uncertainty relations are considered. The corresponding measures of unsharpness are connected for the case where both the measurements are equally unsharp. The connection between the POVM elements and symmetries of the underlying Hamiltonian of the measurement interaction is made explicit and used to perform the joint measurement in arbitrary directions. Finally, in the case of three observables, we derive a necessary condition for the approximate joint measurement and use it to show the relative freedom available when the observables are non-orthogonal.

Manik Banik, Md. Rajjak Gazi, Sibasish Ghosh A +1 more

Bohr's complementarity principle is one of the central concepts in quantum mechanics which restricts joint measurement for certain observables. Of course, later development shows that joint measurement could be possible for such observables with the introduction of a certain degree of unsharpness or fuzziness in the measurement. In this paper, we show that the optimal degree of unsharpness, which guarantees the joint measurement of all possible pairs of dichotomic observables, determines the degree of nonlocality in quantum mechanics as well as in more general no-signaling theories.

Arindam Mitra