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

TL;DR

This paper investigates the entropy of bosonic systems at equilibrium using both empirical and Bayesian methods, unifying classical and quantum perspectives through information theory.

Contribution

It introduces a unified framework for analyzing quantum entropy of bosonic systems by connecting empirical and Bayesian approaches via information-theoretic inequalities.

Findings

The probability distributions converge to the multinomial distribution in the thermodynamic limit.

The physical entropy is identified with the Shannon entropy of occupancy numbers.

Bayesian and empirical methods are integrated into an 'infomechanical' framework.

Abstract

The paper analyzes the probability distribution of the occupancy numbers and the entropy of a system at the equilibrium composed by an arbitrary number of non-interacting bosons. The probability distribution is derived both by tracing out the environment from a bosonic eigenstate of the union of environment and system of interest (the empirical approach) and by tracing out the environment from the mixed state of the union of environment and system of interest (the Bayesian approach). In the thermodynamic limit, the two coincide and are equal to the multinomial distribution. Furthermore, the paper proposes to identify the physical entropy of the bosonic system with the Shannon entropy of the occupancy numbers, fixing certain contradictions that arise in the classical analysis of thermodynamic entropy. Finally, by leveraging an information-theoretic inequality between the entropy of the multinomial distribution and the entropy of the multivariate hypergeometric distribution, Bayesianism and empiricism are integrated into a common ''infomechanical'' framework.