As a preliminary step, it is necessary to focus on some important notions from the usual interpretation of quantum physics, which is predictive about measurement results and retro- dictive about state preparations. These tasks need conditional probabilities linked by Bayes’ theorem, and in the most general case, the expression of probabilities on the Hilbert space H is given by the recent generalization (2003) [9] of Gleason’s theorem [10]. This theorem is only based on very general requirements about the probabilities and the mathematical structure of the Hilbert space, which is the starting point of any quantum de- scription. It states that any system can be described by a density operator . More precisely, when we propose a property P$_{n}$ about a system corresponding to a precise value for a given observable , the proposition about the state of this system is represented by the projector on the eigenstates corresponding to this value. In the most general case, such a proposi- tion can also be represented by a hermitian and positive operator ˆ P$_{n}$ in the Hilbert space. The probability Pr ( n ) of checking this property on the system should satisfy the following conditions: According this theorem for a system needing predictions (i.e. with a Hilbert space of dimension D ≥ 2), this probability is given by Pr ( n ) = Tr { ˆ ρ ˆ P$_{n}$ } in which ˆ ρ is a hermitian, positive, and normalized operator, allowing us to make predictions about any properties of the system. This is the reason why we call this operator the state of the system , and the probabilities are in fact conditioned on this state of knowledge. When we reduce this operator in its eigenbasis, we retrieve a statistical mixture usually used in teaching for introducing the density matrix. Thus, each time that we can write the probabilities concerning any exhaustive set of propositions about the system in this previous form, we can then determine the state of the system thanks to this theorem. 3 II. STATES AND PROPOSITIONS 1. 0 ≤ Pr ( n ) ≤ 1 for any proposition P$_{n}$ . 2. ∑ n Pr ( n ) = 1 for any exhaustive set of propostions such that ∑ n ˆ P$_{n}$ = ˆ 1. 3. Pr ( n$_{1}$ or n$_{2}$ or ... ) = Pr ( n$_{1}$ ) + Pr ( n$_{2}$ ) + ... for any non-exhaustive set of propositions such that ˆ P$_{n}$$_{1}$ + ˆ P$_{n}$$_{2}$ + ... ≤ ˆ 1 .