We study the origin of quantum probabilities as arising from non-boolean propositionaloperational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorvian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).

We present a quantum version of the generalized $(h,\phi)$-entropies, introduced by Salicr\'u \textit{et al.} for the study of classical probability distributions. We establish their basic properties, and show that already known quantum entropies such as von Neumann, and quantum versions of R\'enyi, Tsallis, and unified entropies, constitute particular classes of the present general quantum Salicr\'u form. We exhibit that majorization plays a key role in explaining most of their common features. We give a characterization of the quantum $(h,\phi)$-entropies under the action of quantum operations, and study their properties for composite systems. We apply these generalized entropies to the problem of detection of quantum entanglement, and introduce a discussion on possible generalized conditional entropies as well.Comment: 26 pages, 1 figure. Close to published versio