The Sobolev inequality and the Tsallis entropic uncertainty relation

Steeve Zozor, G. M. Bosyk, M. Portesi

Alexey E. Rastegin

A way to pose the entropic uncertainty principle for trace-preserving super-operators is presented. It is based on the notion of extremal unraveling of a super-operator. For a given input state, different effects of each unraveling result in some probability distribution at the output. As is shown, all Tsallis' entropies of positive order as well as some of Rényi's entropies of this distribution are minimized by the same unraveling of a super-operator. Entropic relations between a state ensemble and the generated density matrix are revisited in terms of both the adopted measures. Using Riesz's theorem, we obtain two uncertainty relations for any pair of generalized resolutions of the identity in terms of the Rényi and Tsallis entropies. The inequality with Rényi's entropies is an improvement of the previous one, whereas the inequality with Tsallis' entropies is a new relation of a general form. The latter formulation is explicitly shown for a pair of complementary observables in a d -level system and for the angle and angular momentum. The derived general relations are immediately applied to extremal unravelings of two super-operators.

Alexey E. Rastegin

An entropic approach to formulating uncertainty relations for the number–annihilation pair is considered. We construct some normal operator that traces the annihilation operator as well as commuting quadratures with a complete system of common eigenfunctions. Expanding the measured wave function with respect to them, one obtains a relevant probability distribution. Another distribution is naturally generated by measuring the number operator. Due to the Riesz–Thorin theorem, there exists a nontrivial inequality between corresponding functionals of the above distributions. We find the bound in this inequality and further derive uncertainty relations in terms of both the Rényi and Tsallis entropies. Entropic uncertainty relations for a continuous distribution as well as relations for a discretized one are presented.

Iwo Białynicki‐Birula

Grzegorz Andrzej Wilk, Z. Włodarczyk

Uncertainty relations emerging from the Tsallis entropy are derived and discussed. In particular we found a positively defined function that saturates the so-called entropic inequalities for entropies characterizing the physical states under consideration.

Masanori Ohya, Dėnes Petz