• Citer

Multidimensional entropic uncertainty relation based on a commutator matrix in position and momentum spaces

par Hertz, Anaëlle ;Van Bever, Luc ;Cerf, Nicolas
Référence Physical Review A, 97, 1, 012111
Publication Publié, 2018-01

Article révisé par les pairs

• Détails
• Contenu

Résumé :

The uncertainty relation for continuous variables due to Byałinicki-Birula and Mycielski [I. Białynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)CMPHAY0010-361610.1007/BF01608825] expresses the complementarity between two n-tuples of canonically conjugate variables (x1,x2,...,xn) and (p1,p2,...,pn) in terms of Shannon differential entropy. Here we consider the generalization to variables that are not canonically conjugate and derive an entropic uncertainty relation expressing the balance between any two n-variable Gaussian projective measurements. The bound on entropies is expressed in terms of the determinant of a matrix of commutators between the measured variables. This uncertainty relation also captures the complementarity between any two incompatible linear canonical transforms, the bound being written in terms of the corresponding symplectic matrices in phase space. Finally, we extend this uncertainty relation to Rényi entropies and also prove a covariance-based uncertainty relation which generalizes the Robertson relation.