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 ρ-bar A-bar obtained by cutting the classical system along the imaginary time (inverse temperature) axis. We show that the von Neumann entropy S-bar ent of ρ-bar A-bar 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 S-bar ent is non-extensive for all temperatures T. For the integrable transverse Ising and XXZ models we demonstrate that the entanglement spectra of ρ-bar A-bar in the limit T → 0 are described by free-fermion Hamiltonians and reduce to those of the regular reduced density matrix ρA—obtained by a spatial instead of an imaginary time cut—up to degeneracies. (paper)