Convex-roof entanglement measures of density matrices block diagonal in disjoint subspaces for the study of thermal states

TL;DR

This paper proves that for certain block-diagonal density matrices, entanglement can be computed as a weighted average of entanglements within each block, simplifying analysis especially for thermal states with symmetries.

Contribution

It introduces a method to calculate entanglement of block-diagonal density matrices as an average over blocks, aiding analysis of thermal states with symmetry.

Findings

Entanglement of block-diagonal states equals the weighted sum of block entanglements.

Method simplifies entanglement calculation for thermal states with Hamiltonian symmetries.

Demonstrated diversity of temperature-dependent entanglement behaviors.

Abstract

We provide a proof that entanglement of any density matrix which block diagonal in subspaces which are disjoint in terms of the Hilbert space of one of the two potentially entangled subsystems can simply be calculated as the weighted average of entanglement present within each block. This is especially useful for thermal-equilibrium states which always inherit the symmetries present in the Hamiltonian, since block-diagonal Hamiltonians are common as are interactions which involve only a single degree of freedom of a greater system. We exemplify our method on a simple Hamiltonian, showing the diversity in possible temperature-dependencies of Gibbs state entanglement which can emerge in different parameter ranges.