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Communications in Mathematical Physics

Uncertainty relations for information entropy in wave mechanics

Iwo Białynicki‐Birula, Jerzy Mycielski

Published Jun 1, 1975• 1,146 citations
Publisher
DOI
10.1007/bf01608825

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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 Robertson–Schrödinger uncertainty relation and 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.
We use the concept of entropy power to introduce a new one-parameter class of information-theoretic uncertainty relations. This class constitutes an infinite hierarchy of uncertainty relations, which allows to determine the shape of the underlying information-distribution function by measuring the relevant entropy powers. The efficiency of such uncertainty relations in quantum mechanics is illustrated with two examples: superpositions of two squeezed states and the Lévy-type heavy-tailed wave function. Improvement over both the variance-based and Shannon entropy based uncertainty relations is demonstrated in both these cases.