Entanglement of Distillation for Lattice Gauge Theories

Abstract

We study the entanglement structure of lattice gauge theories from the local operational point of view, and, similar to Soni and Trivedi (arXiv:1510.07455), 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 LOCC distillable entanglement, which is obtained by depolarizing the local superselection sectors. We demonstrate that the distillable entanglement is zero for pure abelian gauge theories in the weak coupling limit, while it is in general nonzero for the nonabelian 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.Introduction-The concept of quantum entanglement is at the center of current day physics. While it used to be confined to the realms of quantum information theory, it is now appearing in many other fields. It is used for instance to characterize new phases of matter with topological order [1,2], and it also plays a crucial role as guiding principle in the classical simulation of general quantum many body systems [3][4][5]. Furthermore in the context of the holographic principle and quantum gravity there appears to be an intricate connection between entanglement and geometry [6,7]. At the same time entanglement lies at the root of the firewall paradox in the quantum behavior of black holes [8,9].It is hence highly desirable to gain a better understanding of the entanglement structure of quantum field theories. However, as first pointed out in [10], for gauge theories the concept of entanglement is obscured by their intrinsic non-locality. More formally, the Hilbert space for locally distinct regions is not of tensor product form, which renders the usual rules for computing the entanglement not applicable. While a number of tools have been developed to cope with this computational problem [10][11][12][13][14][15][16][17][18], the physical meaning of entanglement in gauge theories was still unclear.In this letter, we put forward the operational view on entanglement: for us, the physical entanglement is identified as the accessible one. That is, the asymptotic number of EPR-pairs E

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

“…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].…”

“…Let i schematically denote such a sector and let p i be the probability for the normal component of the boundary electric flux associated with this sector to arise. Then it was argued in [7,8] that for an Abelian theory,…”

“…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 [7,8], the extended Hilbert space definition was analysed in more detail and it was also shown that the definition does not agree with an operational measure of entanglement which comes from quantum information theory, and which is related to the entanglement that can be extracted from two halves of a bipartite system in the form of Bell pairs, or the entanglement which is available for use in the process of entanglement dilution.…”

“…This puts a limitation on how much total entanglement can be extracted. [7,8] precisely determined just how much of the total entanglement entropy could actually be extracted in entanglement distillation and dilution. Unlike the total entanglement entropy which depends on the extended Hilbert space definition that is used to define it, the extractable part is tied to an operational definition, based on physical measurements, and is independent of this definition.…”