Uncertainty relations expressed by Shannon-like entropies

Abstract

Besides the well-known Shannon entropy, there is a set of Shannon-like entropies which have applications in statistical and quantum physics. These entropies are functions of certain parameters and converge toward Shannon entropy when these parameters approach the value 1. We describe brie®y the most important Shannon-like entropies and present their graphical representations. Their graphs look almost identical, though by superimposing them it appears that they are distinct and characteristic of each Shannon-like entropy. We try to formulate the alternative entropic uncertainty relations by means of the Shannon-like entropies and show that all of them equally well express the uncertainty principle of quantum physics.

The uncertainty relation expressed by means of a new entropic function

“…(iii) By means of this entropic function we have constructed an alternative uncertainty relation for two noncommuting observables. (iv) It appears that this uncertainty relation expresses the uncertainty principle of quantum physics equally as well as the Shannon and Shannon-like entropic uncertainty relations [1,14]. (v) Due to availability of a number of various entropic functions [17], a question arises about the criteria for the choice of the proper one to be used in a particular case of uncertainty relation.…”

“…There are several probabilistic functions of the probability components which satisfy these demands. The most familiar two are: (1) ( ) = − log and (2) ( ) = (1 − ) If we take the sum of the function (1) ( ) for the uncertainty measure (entropy) we have…”

“…In a previous article published in this Journal [1] we formulated the quantum uncertainty relations for two noncommuting observables by means of the Shannon-like entropic functions. In what follows, we use a new entropic function to formulate the uncertainty relation which is mathematically simpler than the Shannon-like entropies.…”

Signatures of quantum fluctuations in the Dicke model by means of Rényi uncertainty

“…We have shown that the description of the quantum phase transition in terms of the entropic uncertainty relation turns out to be more suitable than in terms of the standard variance-based uncertainty relation [32]. The importance of entropic uncertainty measure for fluctuations has recently been emphasized [33].…”

Uncertainty principle for measurable sets and signal recovery in quaternion domains

“…The uncertainty principle plays an important role in signal processing [22,9,27,19,36,29,21,32,8,20,39,37,6,24,23,41,42], and physics [28,16,30,17,18,33,1,7,40,38]. In quantum mechanics an uncertainty principle asserts that one cannot be certain of the position and of the velocity of an electron (or any particle) at the same time.…”