Entanglement in four-dimensional SU(3) gauge theory

Abstract

We investigate the quantum entanglement entropy for the four-dimensional Euclidean SU(3) gauge theory. We present the first non-perturbative calculation of the entropic c-function (C(l)) of SU(3) gauge theory in lattice Monte Carlo simulation using the replica method. For 0 l 0.7 fm, where l is the length of the subspace, the entropic c-function is almost constant, indicating conformally invariant dynamics. The value of the constant agrees with that perturbatively obtained from free gluons, with 20 % discrepancy. When l is close to the Hadronic scale, the entropic c-function decreases smoothly, and it is consistent with zero within error bars at l 0.9 fm.Quantum entanglement is a fascinating phenomenon that was first highlighted by the Einstein-Podolsky-Rosen paradox [1] and has remained a focus of research activity for decades. If there is a system in a pure quantum state, measurements on a subsystem A determine the results of measurements on its complement B, even if no causal communication is possible between the two measurements. The entanglement entropy S A of subsystem A is defined as von Neumann entropy corresponding to the reduced density matrix ρ A :where ρ A = Tr HB [ρ tot ], and it is assumed that the total Hilbert space is a direct product of two subspaces corresponding to the subsystems considered, H tot = H A ⊗H B . More generally, studies of the entanglement entropy become central in cases of complex systems with strong interactions, where the properties of the ground state cannot be evaluated directly. In particular, the notion of quantum entanglement is crucial for the theory of quantum phase transitions, i.e., non-thermal phase transitions at temperature T = 0 [2][3][4]. In physics of black holes, consideration of the quantum entanglement is central to discussions of the information paradox [5], which challenges the consistency of general relativity and quantum mechanics.Applications to field theory are more recent. First of all, the entanglement entropy is ultraviolet divergent in field theory [6]. In more detail, one considers the vacuum state and defines the subsystem A as a slab of length l in one of the spatial dimensions, at a fixed time slice. Then, the entanglement entropy contains, as its most divergent term, a term that is proportional to |∂A|/a d−1 , where d is the number of spatial dimensions, a is the lattice spacing, and |∂A| is the area of the boundary surface between the slab and the rest of the space. To eliminate this divergence, which depends on details of the UV cutoff, one focuses on the entropic c-function [7]:

Logarithmic black hole entropy corrections and holographic Rényi entropy

“…We found that, for fixed C, as we 11 For d = 2, the expression for renormalized entanglement entropy is similar to the entropic c-function [51]. The fact that c-function decreases monotonically from UV to IR has also been confirmed both in lattice as well as in holographic studies [52,53]. increase the number of spacetime dimensions the magnitude of the ratio S q /S 1 increases.…”

Holographic anisotropic background with confinement-deconfinement phase transition

“…The entan-glement entropy of the selected domain A is defined as the von Neumann entropy of the reduced density matrix obtained by tracing out the degrees of freedom located out of domain A. The entanglement entropy is hard to compute in QCD, but it is suitable to compute it in the lattice [23][24][25], see also [26]. The entanglement entropy can be also evaluated at the gravity side.…”

“…25. The "scaling factors" m SF x,y (z, z h , ν) and m EF x,y (z, z h , ν) defined by (5.22) and (5.23) and their UV approximations (5.20) and (5.21) shown by the dashed lines.The scaling functions m SF x,y and m EF x,y shown in(25) are defined by the formula similar to (5.18) and(5.19)…”

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.…”