We develop a possibilistic semantic for quantum phenomena in an operational perspective. This semantic is based on a Chu duality between preparation processes and yes/no tests, the target space being a three-valued set equipped with an informational interpretation. After having defined the notion of states, we develop a precise axiomatic for the space of states. The 'information principle', proposed as a building block for some quantum axiomatic program, is translated into our framework to emphasize the quantum character of our description. This principle suffices to constrain the space of states to be a locally-boolean qualitative domain. The subset of pure states is then characterized within this domain structure. After having carefully precised the notions of properties and measurements, we explore the notion of compatibility between measurements. The existence of minimally-disturbing measurements is then emphasized as a key axiom to describe the space of yes/no tests. The conditions of existence of such measurements is translated into a simple property on the space of states, and an explicit formula for these measurements is given. Having reduced the space of yes/no tests to this set of minimally-disturbing operations, our Chu space becomes bi-extensional. The space of 'descriptions', associated to families of compatible measurements, inherits a structure of coherence domain. A subset of properties corresponding to 'classical properties' is identified and explored. Endly, the symmetries of the system are characterized as a general sub-class of Chu morphisms. We prove that these symmetries preserve the class of minimally-disturbing measurements and the orthogonality relation between states. Explicit expressions of these symmetries are identified for a large class of them, called 'linear symmetries'.