Role of information theoretic uncertainty relations in quantum theory

Abstract

Uncertainty relations based on information theory for both discrete and continuous distribution functions are briefly reviewed. We extend these results to account for (differential) Rényi entropy and its related entropy power. This allows us to find a new class of information-theoretic uncertainty relations (ITURs). The potency of such uncertainty relations in quantum mechanics is illustrated with a simple two-energy-level model where they outperform both the usual RobertsonSchrödinger uncertainty relation and Kraus-Maassen Shannon entropy based uncertainty relation. In the continuous case the ensuing entropy power uncertainty relations are discussed in the context of heavy tailed wave functions and Schrödinger cat states. Again, improvement over both the Robertson-Schrödinger uncertainty principle and Shannon ITUR is demonstrated in these cases. Further salient issues such as the proof of a generalized entropy power inequality and a geometric picture of information-theoretic uncertainty relations are also discussed.

Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

“…The dominant contribution here is given by I 3 . Substituting asymptotics (44) in I 3 , making the change of variable t := x 4n and using ( [67], Lemma 2.1) we arrive to…”

“…For a revision of their properties see [38,39,40,41,42,43,44] and the reviews [45,46]. The Rényi entropies and their associated uncertainty relations have been widely used to investigate a great deal of quantum-mechanical properties and phenomena of physical systems and processes [29,45,46,44], ranging from the quantum-classical correspondence [47] and quantum entanglement [48] to pattern formation and Brown processes [49,50], fractality and chaotic systems [51,52], quantum phase transition [53] and disordered systems [54]. Moreover, the knowledge of these quantities allows us to reconstruct the corresponding probability density under certain conditions [41,55].…”

One-parameter class of uncertainty relations based on entropy power

“…This type of expression was studied in [18,19] where it was shown that the only class of solution of (5) is…”

Information Entropy of Nanostructure systems

“…According to Density Function theory, a many fermion system can be completely characterized by the single particle distribution density, which is is that an increase of S(mom) corresponds to a decrease of S(pos) and vice-versa, which indicates that a diffuse density distribution ρ(p) in momentum space is associated with a localized density distribution ρ(r) in configuration space [3][4][5][6][7][8][9][10][11][12]. In this paper, we investigate the use of the position space information entropy as an indicator of the entanglement for nanostructure system.…”