Entropy and the Uncertainty Principle

A minimum principle is obtained for the sum of entropies of two distributions related as the absolute squares of a Fourier transform pair. The minimum is shown to be attained for a Gaussian pair. The joint entropy is calculated for two other Fourier pairs of interest. Applications to the uncertainty principle are made by defining a joint entropy for position and momentum. A generalized uncertainty principle, for any set of observables not simultaneously measurable, is conjectured.