Interpretation of Uncertainty Relations for Three or More Observables

Conventional quantum uncertainty relations (URs) contain dispersions of two observables. Generalized URs are known which contain three or more dispersions. They are derived here starting with suitable generalized Cauchy inequalities. It is shown what new information the generalized URs provide. Similar interpretation is given to generalized Cauchy inequalities.Comment: 6 page

The Convex Closure of the Output Entropy of Infinite Dimensional Channels and the Additivity Problem

The continuity properties of the convex closure of the output entropy of infinite dimensional channels and their applications to the additivity problem are considered. The main result of this paper is the statement that the superadditivity of the convex closure of the output entropy for all finite dimensional channels implies the superadditivity of the convex closure of the output entropy for all infinite dimensional channels, which provides the analogous statements for the strong superadditivity of the EoF and for the additivity of the minimal output entropy. The above result also provides infinite dimensional generalization of Shor's theorem stated equivalence of different additivity properties. The superadditivity of the convex closure of the output entropy (and hence the additivity of the minimal output entropy) for two infinite dimensional channels with one of them a direct sum of noiseless and entanglement-breaking channels are derived from the corresponding finite dimensional results. In the context of the additivity problem some observations concerning complementary infinite dimensional channels are considered.Comment: 24 page