Complementarity enforced by joint measurability of unsharp observables

Guruprasad Kar

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.

Paul Busch, C. R. Shilladay

The fundamental principles of complementarity and uncertainty are shown to be related to the possibility of joint unsharp measurements of pairs of noncommuting quantum observables. A joint measurement scheme for complementary observables is proposed. The measured observables are represented as positive operator valued measures (POVMs), whose intrinsic fuzziness parameters are found to satisfy an intriguing pay-off relation reflecting the complementarity. At the same time, this relation represents an instance of a Heisenberg uncertainty relation for measurement imprecisions. A model-independent consideration shows that this uncertainty relation is logically connected with the joint measurability of the POVMs in question.

Arindam Mitra

Paul Busch

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.