Equiprobability, inference, and entropy in quantum theory

An inference procedure is required to assign a density operator D to a system when the only available information is the set of expectation values a i of some observables A i , commuting or not. To do this, we consider as a single “supersystem” the Gibbsian ensemble consisting of N replicas of the system, on which thought experiments compatible with the data a i can be performed. In this supersystem, we identify the expectation values a i with mean values, and implement the principle that equal ignorance should be represented by equiprobability. New features are brought in through the non-commutativity of the observables A i , and through the operator nature of the quantum states. These generate difficulties, both conceptual and technical, which we discuss and resolve. The density operator D of the system considered is obtained from that of the supersystem by taking the partial trace over the N − 1 other systems. The rather involved calculations are accomplished by field theoretical techniques. They provide a generalized canonical distribution, with deviations which vanish as N → ∞, the same result as would be obtained by maximizing von Neumann's entropy with the constraints 〈 A i 〉 = a i . We succeed thus to justify the quantal maximum entropy principle and to construct simultaneously von Neumann's entropy, starting from the idea of equiprobability in the supersystem.