Some entropic extensions of the uncertainty principle

STEEVE ZOZOR; MARIELA PORTESI; CHRISTOPHE VIGNAT

Toronto

Simposio; IEEE ISIT 1676-1680 (2008). (ISIT 2008: IEEE International Symposium on Information Theory); 2008

IEEE

In connection with the uncertainty principle in quantum mechanics (Heisenberg) or in time-frequency analysis (Heisenberg-Gabor), we study its formulation in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [1] and Zozor et al. [2]. These results can be considered as generalizations of the Heisenberg inequalities in the sense that they measure the mutual uncertainty of a random variable (or wave function) and its conjugated random variable (or Fourier transformed wave function) through their associated Renyi entropies with conjugated indexes. We consider here the more general case where the entropic indexes are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indexes alpha and beta in the plane (alpha, beta). Our results explain and extend a recent study by Luis [3], where states with quantum fluctuations below the Gaussian case are discussed at the single point (2, 2).