On complete and incomplete sets of observables, the principle of maximum entropy—revisited

R. S. Ingarden is one of the founders of the principle of maximum entropy (formulated independently by E. T. Jaynes), an important approach to basic assumptions of statistical physics. He is primarily responsible for recognizing this principle and fostering its developoment (cf. e.g. [1–3]). In this paper we discuss some questions connected, in natural way, with the principle of maximum entropy, namely, the question of the minimal number η of observables Q1,…,Qη, whose mean values at some instants t1,…,ts determine the statistical state of an N-level open quantum system (we assume that the time evolution of the system in question is given by a semigroup of linear transformations with a generator L) and a similar problem of the minimal number κ of instant t1,…,tκ which are necessary for reconstructibility of trajectories of the system. To give an exact answer to the above questions we introduce the following two concepts: index of cyclicity of a given generator L of time evolution and index of reconstructibility (for a given generator L and a given set of observations Q1,…,Qn).