Remarks on uncertainties relations in quantum physics and beyond it

Seth Lloyd, Vittorio Giovannetti, Lorenzo Maccone

John Maddox

Aikaterini Mandilara, Nicolas J. Cerf

We rederive the Schr\"odinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation.

Peter Harremöes D, Petr Jizba B, G. Kaniadakis +1 more

Yunlong Xiao, Naihuan Jing

The uncertainty principle is one of the characteristic properties of quantum theory based on incompatibility. Apart from the incompatible relation of quantum states, mutually exclusiveness is another remarkable phenomenon in the information- theoretic foundation of quantum theory. We investigate the role of mutual exclusive physical states in the recent work of stronger uncertainty relations for all incompatible observables by Mccone and Pati and generalize the weighted uncertainty relation to the product form as well as their multi-observable analogues. The new bounds capture both incompatibility and mutually exclusiveness, and are tighter compared with the existing bounds.

D. A. Trifonov

Three remarks concerning the form and the range of validity of the state-extended characteristic uncertainty relations (URs) are presented. A more general definition of the uncertainty matrix for pure and mixed states is suggested. Some new URs are provided.