Entanglement of Distillation for Lattice Gauge Theories

We study the entanglement structure of lattice gauge theories from the local operational point of view, and, similar to Soni and Trivedi [J. High Energy Phys. 1 (2016) 1], we show that the usual entanglement entropy for a spatial bipartition can be written as the sum of an undistillable gauge part and of another part corresponding to the local operations and classical communication distillable entanglement, which is obtained by depolarizing the local superselection sectors. We demonstrate that the distillable entanglement is zero for pure Abelian gauge theories at zero gauge coupling, while it is in general nonzero for the non-Abelian case. We also consider gauge theories with matter, and show in a perturbative approach how area laws—including a topological correction—emerge for the distillable entanglement. Finally, we also discuss the entanglement entropy of gauge fixed states and show that it has no relation to the physical distillable entropy.

Figure 1

(a) The setting of lattice gauge theory: the action of a gauge transformation $U_v$ is determined by the orientation of the edges. (b) A bipartite cut: the edges along the cut are labeled by irreps $r_1,\dots ,r_6$ of the group action at the six different boundary vertices (thick points), giving rise to a direct sum structure. Fixing theses irreps produces a direct product structure on each boundary edge, with one factor lying on the inside and one factor on the outside.

Figure 2

Conjectured phase diagram for the distillable entanglement in the ${\mathbb{Z}}_2$ lattice gauge theory: $h_z$ ($h_x$) are the coupling constants for the hopping term between matter degrees of freedom and for the ${\sigma }_x$ perturbation on the gauge fields, c.f. Refs. [45, 46]. Along the $h_z=0$ axis (red shaded), $E_D^{\text{gauge}}$ for any compact region $A$ is zero, while it satisfies $E_D^{\text{gauge}}=c|A|-\gamma$, with $\gamma >0$ nonzero topological correction for $h_z>0$ and $h_x$ not too big (region I). Outside of region I, we have $\gamma =0$ and also $E_D^{\text{gauge}}\approx 0$ for $h_z$, $h_x$ large, and the regions II and III are equivalent.