Observables and Uncertainty

Jinchuan Hou, Kan He

Uncertainty relations describe the lower bound of product of standard deviations of observables. By revealing a connection between standard deviations of quantum observables and numerical radius of operators, we establish a universal uncertainty relation for any $k$ observables, of which the formulation depends on the even or odd quality of $k$. This universal uncertainty relation is tight at least for the cases $k=2n$ and $k=3$. For two observables, the uncertainty relation is exactly a simpler reformulation of Schrödinger's uncertainty principle.

Minyi Huang

Uncertainty is a fundamental and important concept in quantum mechanics. In this work, using the technique in matrix theory, we propose an uncertainty relation of four observables and show that the uncertainty constant is tight. It is argued that this method can deal with the several known uncertainty relations for two, three and four observables in a unified way. The result is also compared with other uncertainty relations of four observables.

Günter Ludyk

Hui-Hui Qin, Shao-Ming Fei, Xianqing Li-Jost

We investigate the product form uncertainty relations of variances for n (n ≥ 3) quantum observables. In particular, tight uncertainty relations satisfied by three observables has been derived, which is shown to be better than the ones derived from the strengthened Heisenberg and the generalized Schrödinger uncertainty relations, and some existing uncertainty relation for three spin-half operators. Uncertainty relation of arbitrary number of observables is also derived. As an example, the uncertainty relation satisfied by the eight Gell-Mann matrices is presented.