The determination of bounds of the β-entropic sum of two noncommuting observables

This article deals with the Havrda-Charvat β-entropy as a dispersion measure for the quantum observables. This entropy depends on a parameter β and thus represents a set of uncertainty measures which includes the Shannon entropy as a special case. However, for certain values of β, it is considerably simpler to handle mathematically than the Shannon entropy. It is possible, e.g. to find easier the bounds of the sum of β-entropies for two noncommuting observables than the bounds of the sum of their Shannon entropies. By means of the variational method we determine such a probability distribution of noncommuting observables which minimizes or maximizes their β-entropic sum. As an example, we compute the bounds of β-entropic sum of the spin components of a spin-12 particle for the arbitrary β, and two selected values of β, Putting β → 1 in the β-entropic sum we get the upper and lower bounds of their Shannon entropy sum.