Holographic Entanglement Entropy by Mukund Rangamani & Tadashi Takayanagi

Author:Mukund Rangamani & Tadashi Takayanagi
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

While we have derived the first law for entanglement entropy (8.1.11) by examining small subsystems in a holographic setting, let us now see how a general statement of the entanglement first law follows from the relative entropy in any QFT [56]. We have already indicated some of this in our discussion in Sect. 2.​5, but it is instructive to re-examine those statements in light of the current discussion.

The first law of entanglement follows from the stationarity of relative entropy for perturbation about a reference state (2.​5.​8). It relates the linear change in the modular Hamiltonian to the change in the entanglement entropy. One way to read the expression is that a state ρ 0 is indistinguishable from another ρ 0 +ε ρ 1 to by examining the entanglement alone. In our discussion of Sect. 8.1.1, we picked an excited state, which one would expect to be clearly distinguishable from the ground state. It certainly would be if we focused on the macroscopic features of the excitation. Should we however only ask questions about relatively small subregions and the reduced density matrix induced thereon, then we are restricting attention to a very small part of the information available. The reduced density matrix for the excited state ∣​E〉 is almost indistinguishable from that of the vacuum as long as we satisfy the condition outlined (8.1.4). Within this limit, one is guaranteed to find

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