Abstract
We propose a definition for the entanglement entropy of a gauge theory on a spatial lattice. Our definition applies to any subset of links in the lattice, and is valid for both Abelian and Non-Abelian gauge theories. For $\mathbb{Z}_N$ and $U(1)$ theories, without matter, our definition agrees with a particular case of the definition given by Casini, Huerta and Rosabal. We also argue that in general, both for Abelian and Non-Abelian theories, our definition agrees with the entanglement entropy calculated using a definition of the replica trick. Our definition, however, does not agree with some standard ways to measure entanglement, like the number of Bell pairs which can be produced by entanglement distillation.
Highlights
Of electric or magnetic flux, which are non-local
We propose a definition for the entanglement entropy of a gauge theory on a spatial lattice
We see that our definition yields an expression for the entanglement entropy which is similar in form to that obtained in the Abelian cases discussed earlier
Summary
Our starting point is a Z2 lattice gauge theory without matter, one of the simplest examples of a gauge theory. Consider a gauge-invariant state, |ψ , and some subset of all the spatial links. We will call these links as lying in the “inside”, our discussion is general and does not require the collection of links to form a closed or even connected region. As was discussed in the introduction there is an extended Hilbert space, H, which is obtained in this case by taking the tensor product of all the two dimensional Hilbert spaces defined at each link. This Hilbert space by definition admits a tensor product decomposition in terms of the Hilbert spaces of links lying inside, Hin, and outside (i.e. the rest of the links), Hout. In the context of an extended lattice construction, as we will discuss further in section 6.3, the same definition was given, both for the Abelian and Non-Abelian cases, in1 [12, 14]
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