A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics

TL;DR

This paper introduces a differential-geometric framework for quantum ignorance, defining a volume on the manifold of purifications, and demonstrates how systems evolve towards equilibrium states of maximal volume and entanglement, aligning with thermodynamic principles.

Contribution

It develops a novel geometric approach to quantum ignorance using volume measures on purification manifolds, linking entropic properties with coarse-graining in quantum systems.

Findings

System states evolve towards maximum volume equilibrium states.

Volume correlates with von Neumann entropy, being zero for pure states.

Equilibrium states dominate the coarse-grained space in large systems.

Abstract

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator $\rho_S$. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of $\rho_S$. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from $\rho_S$. Using examples where the SOI are generated using representations of $SU(2)$, $SO(3)$, and $SO(N)$, we show two features of the CG. (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grainied space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to maximum entanglement between system and environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function w.r.t the purity of $\rho_S$. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG.