Maximum Geometric Quantum Entropy

TL;DR

This work reframes the one-to-many relation between a density matrix $ρ$ and its pure-state ensembles as an inference problem. By introducing the Maximum Geometric Entropy Principle (MaxGEP) at fixed quantum information dimension $𝔇$ and using Geometric Quantum Entropy $h_{𝔇}$, it derives principled ensemble selections behind $ρ$, with analytic solutions for $𝔇=0$ and $𝔇=2(d_S-1)$. The results connect to the Hughston–Josza–Wootters theorem and geometric quantum mechanics, and they show how ensembles can emerge from environmental conditioning, time averaging, or isolated dynamics. This provides a scalable, information-theoretic method to quantify and select ensembles beyond the eigenensemble, with implications for quantum thermodynamics and quantum state reconstruction. The framework clarifies how geometry, information dimension, and dynamics jointly shape the feasible ensembles compatible with a given $ρ$.

Abstract

Any given density matrix can be represented as an infinite number of ensembles of pure states. This leads to the natural question of how to uniquely select one out of the many, apparently equally suitable, possibilities. Following Jaynes' information-theoretic perspective, this can be framed as an inference problem. We propose the Maximum Geometric Quantum Entropy Principle to exploit the notions of Quantum Information Dimension and Geometric Quantum Entropy. These allow us to quantify the entropy of fully arbitrary ensembles and select the one that maximizes it. After formulating the principle mathematically, we give the analytical solution to the maximization problem in a number of cases and discuss the physical mechanism behind the emergence of such maximum entropy ensembles.

Theorems & Definitions (1)

• Proposition 1: Maximum Geometric Entropy Principle