Uncertainty Principle and Relativity

Jos Uffink, Jan Hilgevoord

Pavle Ilija Premović

Jan Hilgevoord, Jos Uffink

The uncertainty principle of position and momentum is a mathematically more involved relation than is usually realised. The well-known Heisenberg uncertainty relation between standard deviations is much too weak to express the uncertainty principle. By employing other notions of width, relations are given which do express the uncertainty principle.

Ronny Machluf

The phenomenon in the essence of classical uncertainty principles is well known since the thirties of the last century. We introduce a new phenomenon which is in the essence of a new notion that we introduce: "Generalized Uncertainty Principles". We show the relation between classical uncertainty principles and generalized uncertainty principles. We generalized "Landau-Pollak-Slepian" uncertainty principle. Our generalization relates the following two quantities and two scaling parameters: 1) The weighted time spreading $\int_{-\infty}^\infty |f(x)|^2w_1(x)dx$, ($w_1(x)$ is a non-negative function). 2) The weighted frequency spreading $\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω$. 3) The time weight scale $a$, ${w_1}_a(x)=w_1(xa^{-1})$ and 4) The frequency weight scale $b$, ${w_2}_b(ω)=w_2(ωb^{-1})$. "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights $w_1(x)$ and $w_2(ω)$, we introduced a three dimensional set in $R^3$ that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights, $w_1(x)=w_2(x)=x^k$, $k\in N$, up to a constant.

Sibel Başkal, Young Suh Kim, Marilyn E. Noz

Otto Charles Anthony Gadea, Gardo Blado

We apply the generalised uncertainty principle (GUP) to the entropic uncertainty relation conditions on quantum entanglement. In particular, we study the GUP corrections to the Shannon entropic uncertainty condition for entanglement. We combine previous work on the Shannon entropy entanglement criterion for bipartite systems and the GUP corrections to the Shannon entropy for a single system to calculate the GUP correction for an entangled bipartite system. As in an earlier paper of the second author, which dealt with variance relations, it is shown that there is an increase in the upper bound for the entanglement condition upon the application of the generalised uncertainty principle. Necessary fundamental concepts of the generalised uncertainty principle, entanglement and the entropic uncertainty relations are also discussed. This paper puts together the concepts of entanglement, entropic uncertainty relations and the generalised uncertainty principle all of which have been separately discussed in pedagogical papers by Schroeder, Majernik et al., Blado et al. and Sprenger.