Entropy of the Canonical Occupancy (Macro) State in the Quantum Measurement Theory

Abstract

The paper analyzes the entropy of a system composed by an arbitrary number of indistinguishable particles at the equilibrium, defining entropy as a function of the quantum state of the system, not of its phase space representation. Our crucial observation is that the entropy of the system is the Shannon entropy of the random occupancy numbers of the quantum states allowed to system's particles. We consider the information-theoretic approach, which is based on Jaynes' maximum entropy principle, and the empirical approach, which leads to canonical typicality in modern quantum thermodynamics. In the information-theoretic approach, the occupancy numbers of particles' quantum states are multinomially distributed, while in the empirical approach their distribution is multivariate hypergeometric. As the number of samples of the empirical probability tends to infinity, the multivariate hypergeometric distribution tends to the multinomial distribution. This reconciles, at least in the limit, the two approaches. When regarded from the perspective of quantum measurement, our analysis suggests the existence of another kind of subjectivism than the well-known subjectivism that characterizes the maximum entropy approach. This form of subjectivity is responsible for the collapse of entropy to zero after the quantum measurement, both in the information-theoretic and in the empirical approaches