Möbius operators and non-additive quantum probabilities in the Birkhoff–von Neumann lattice

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 Möbius (or non-additivity) operators which are defined through Möbius transforms, and which are shown to be intimately related to commutators. The lack of distributivity in the Birkhoff–von Neumann lattice ΛdΛd, 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 ΛdΛd, 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 Möbius 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.