A contour for the entanglement negativity of bosonic Gaussian states

TL;DR

This paper introduces a contour function for entanglement negativity in bosonic Gaussian states, analyzing its divergence properties and behavior in one- and two-dimensional models, with numerical and analytical insights.

Contribution

It constructs a novel contour function for entanglement negativity and moments of partial transpose, providing detailed divergence analysis and numerical results for harmonic chains.

Findings

Contour diverges only at entangling points

Logarithmic negativity derivative decreases monotonically

Analytic expressions describe divergence behavior

Abstract

We construct a contour function for the logarithmic negativity and the logarithm of the moments of the partial transpose of the reduced density matrix for multimode bosonic Gaussian states of a free lattice model. In one spatial dimension, numerical results are obtained for harmonic chains either in the ground state or at finite temperature, by considering, respectively, either a subsystem made by two adjacent or disjoint blocks on the line or a bipartition of the circle. The contour function of the logarithmic negativity diverges only at the entangling points, while the contour function for the logarithm of the moments of the partial transpose is divergent also at the boundary of the bipartite subsystem, as functions of the position. In a two-dimensional conformal field theory, analytic expressions that describe these divergencies are discussed. In one spatial dimension, we explore the partial derivative of the logarithmic negativity of two adjacent intervals with respect to the logarithm of the harmonic ratio of their lengths while their ratio and the other parameters are kept fixed. Considering the ground state of the harmonic chain on the line and in the massive regime, we report numerical results showing that this quantity displays a monotonically decreasing behaviour.