Entanglement measures and the quantum to classical mapping

A quantum model can be mapped to a classical model in one higher dimension. Here we introduce a finite-temperature correlation measure based on a reduced density matrix $\bar\rho_{\bar A}$ obtained by cutting the classical system along the imaginary time (inverse temperature) axis. We show that the von-Neumann entropy $\bar S_{\rm ent}$ of $\bar\rho_{\bar A}$ shares many properties with the mutual information, yet is based on a simpler geometry and is thus easier to calculate. For one-dimensional quantum systems in the thermodynamic limit we prove that $\bar S_{\rm ent}$ is non-extensive for all temperatures $T$. For the integrable transverse Ising and $XXZ$ models we demonstrate that the entanglement spectra of $\bar\rho_{\bar A}$ in the limit $T\to 0$ are described by free-fermion Hamiltonians and reduce to those of the regular reduced density matrix $\rho_A$---obtained by a spatial instead of an imaginary-time cut---up to degeneracies.