Commutation of Observables

Arno Böhm

Abraham Lozada, Sarrazin, G. de, P. L. Torres

In this paper, some general results about the concepts of complete sets of commuting observables (CSCO) and irreducible sets of observables (ISO) are obtained. It is proved the following: (i) Any relevant observable is an essential part of some CSCO. (ii) Any relevant observable, which is a CSCO, is an essential part of some ISO. (iii) Let {b α } α ∈ I be a CSCO, or an ISO, then, I is countable

Sneha Munshi, A. K. Pan

Based on the optimal quantum violation of the suitable Bell's inequality, device-independent self-testing of states and observables has been reported. It is well studied that locally commuting or compatible observables cannot be used to reveal quantum nonlocality. Therefore, self-testing of commuting local observables is not possible through the Bell test. In this work, we demonstrate the self-testing of a set of mutually commuting local observables. We show that the optimal quantum violations of suitably formulated bilocality and $n$-locality inequalities in networks uniquely fix the observables of one party to be mutually commuting. In particular, we first demonstrate that in a two-input arbitrary-party star network, two commuting local observables can be self-tested. Further, by considering an arbitrary-input three-party bilocal network scenario, we demonstrate the self-testing of an unbounded number of mutually commuting local observables.

Saptak Bhattacharya

Abstract We establish new uncertainty bounds for commutators and anti-commutators of observables in both finite- and infinite-dimensional quantum systems. Beginning with a short, elementary proof of Robertson’s standard uncertainty principle, we then derive inequalities whose lower bounds remain nonzero, regardless of the number of observables. The main result is a novel trace inequality comparing the block matrices of commutators and anti-commutators of several observables.