On the definition of entanglement entropy in lattice gauge theories

Abstract

We focus on the issue of proper definition of entanglement entropy in lattice gauge theories, and examine a naive definition where gauge invariant states are viewed as elements of an extended Hilbert space which contains gauge non-invariant states as well. Working in the extended Hilbert space, we can define entanglement entropy associated with an arbitrary subset of links, not only for abelian but also for non-abelian theories. We then derive the associated replica formula. We also discuss the issue of gauge invariance of the entanglement entropy. In the Z N gauge theories in arbitrary space dimensions, we show that all the standard properties of the entanglement entropy, e.g. the strong subadditivity, hold in our definition. We study the entanglement entropy for special states, including the topological states for the Z N gauge theories in arbitrary dimensions. We discuss relations of our definition to other proposals.

Entanglement entropy in (3 + 1)-d free U(1) gauge theory

“…For a spatial region which is the inside of a sphere, S 2 , we obtain from the replica trick path integral for the U(1) theory, by a standard argument, the coefficient C. It is given by C = − 31 90 (1. 2) and agrees with the A anomaly coefficient in the theory. Next we analyse how much of the entanglement is extractable through the processes of entanglement distillation and dilution in this system.…”

“…It has been noted earlier that this definition agrees with the replica trick method of calculating the entanglement [1,2]. It has also been noted [7,8] that the extended Hilbert space definition differs from the extractable entanglement, which is the maximum number of Bell pairs that can be obtained in entanglement distillation or available for entanglement dilution, and the difference between the two was precisely stated in [7,8].…”

“…Besides the references above, some key references in the discussion of gauge theory entanglement are [1][2][3][4][5][6][7][8][12][13][14][15][16][17][18][19][20].…”

“…In [1] and [2], a definition, called the Extended Hilbert Space definition, was given for a gauge theory on a spatial lattice, by embedding the gauge-invariant states in a larger Hilbert space which now admits a tensor product product decomposition between the inside and the outside regions. See also earlier important work in [3][4][5][6].…”

Non-conformal entanglement entropy

“…There has been considerable progress in understanding the computation of entanglement entropy in lattice gauge theories, see for example [32,33,[37][38][39], and it would be interesting to explore how the continuum limit of such computations can be matched with our definition of renormalized entanglement entropy.…”