If I ever say anything incorrect, please promptly correct me! The state of a system in classical mechanics is specified by point in phase space, the point giving us the position and velocity at a given instance. Could we rephrase it by saying a vector in phase space specifies the system? If so, it would seem to make transition to QM slightly more natural. My next question is, what is the motivation for thinking that a vector in a Hilbert space represents the state of a quantum? I would appreciate an explanation or a reference to some source that nicely answers this question. My last question, related to the second, is, why is "The situation of two independent observers conducting measurements on a joint quantum system...usually modelled using a Hilbert space of tensor product form, each factor associated to one observer"? Again, is there motivation for this idea?