  The properties of quantum probabilities are linked to the geometry of quantum
mechanics, described by the Birkhoff-von Neumann lattice. Quantum probabilities
violate the additivity property of Kolmogorov probabilities, and they are
interpreted as Dempster-Shafer probabilities. Deviations from the additivity
property are quantified with the Mobius (or non-additivity) operators which are
defined through Mobius transforms, and which are shown to be intimately related
to commutators. The lack of distributivity in the Birkhoff-von Neumann lattice
Lambda , causes deviations from the law of the total probability (which is
central in Kolmogorov's probability theory). Projectors which quantify the lack
of distributivity in Lambda , and also deviations from the law of the total
probability, are introduced. All these operators, are observables and they can
be measured experimentally. Constraints for the Mobius operators, which are
based on the properties of the Birkhoff-von Neumann lattice (which in the case
of finite quantum systems is a modular lattice), are derived.Application of
this formalism in the context of coherent states, generalizes coherence to
multi-dimensional structures.
