# Reeh-Schlieder theorem and the replica trick to calculate entropy

I have recently started learning about algebraic aspects of quantum field theory mainly using Witten's review (https://arxiv.org/abs/1803.04993) and Haag's Local Quantum Physics book. I am not able to understand how the implications from Reeh-Schlieder theorem are compatible with the replica trick used to calculate entropy in quantum field theories, for example the Cardy-Calebrese formalism (https://arxiv.org/abs/0905.4013, https://arxiv.org/abs/hep-th/0405152) to calculate von Neumann entropy of a region. In particular, Reeh-Schlieder theorem implies that we cannot single out degrees of freedom of a given region from its complement on a Cauchy slice (the reason being that there does not exist a Hilbert space factorization into degrees of freedom of the region and that of its complement), but this is an essential part of the Cardy-Calebrese formalism, or the replica trick in general. I maybe misunderstanding something but aren't these two in conflict with each other?

There are some nice discussions/questions on aspects of Reeh-Sclieder theorem and entropy calculations using replica trick:

Intuition for when the replica trick should work and why it works

The Reeh-Schlieder theorem and quantum geometry

Why is entanglement entropy in QFT infinite?

Local algebra of AQFT vs Bisognano Wichmann Theorem

The Role of Rigor

Defining the modular Hamiltonian

The last of the above discussions come closest to the question I am trying to ask, but these don't address my question completely. When we consider the theory on a lattice the nature of operator algebra is different from that in the continuum theory (and this allows for a Hilbert space factorization is a heuristic sense), but how do we guarantee that the continuum limit of the discrete theory will take us to the right continuum operator algebra where there is no Hilbert space factorization? Any explanation about whether or not I am missing something in understanding the two notions would be helpful.

• It's been quite a while since I read about this subject in detail, but I believe the compatibility lies in the fact that the entanglement entropy will in general diverge. See for instance (160) in arxiv.org/abs/1801.10352. I feel that these divergences reflect the fact that rigorously the factorization does not exist. By the way, I would appreciate it if someone more familiar with the subject could confirm or correct this statement.
– Gold
Jun 19, 2022 at 15:36
• So is it correct to say that we can actually not calculate the entropy without resorting to an "approximate" argument where we assume Hilbert space factorization? And this comes with a price of divergent entropy. Witten's recent talks do claim that there is no entropy in the type III von Neumann algebras and these are the correct local algebras in QFT. Jun 19, 2022 at 15:58
• I guess the correct thing to say is that we need one regularization. When the regulator is removed by taking a limit the entanglement entropy diverges. One then needs to renormalize by means of a subtraction. In particular one can do so by defining one renormalized entropy in some state to be the entanglement entropy in that state minus the corresponding one in the vacuum state. If I recall correctly this subtraction removes the divergence. I don't know however how this evades Witten's AQFT issues with the factorization.
– Gold
Jun 19, 2022 at 18:29
• See also arxiv.org/abs/1302.1878 for another very interesting discussion of renormalization of the entanglement entropy in QFT.
– Gold
Jun 19, 2022 at 18:30
• @Gold Thanks for the references. I also found these articles : arxiv.org/abs/1303.0688 and arxiv.org/abs/1301.1300, which talk about entanglement entropy without using the notion of partial trace (although they are not in the context of spacetime regions). They invoke GNS representations and use restriction of (algebraic) states to sub-algebras in order to calculate entanglement entropy, which seems like an interesting way of calculating the entropy. I am not sure if this formalism has been or can be extended to entropy calculation of spacetime regions. Jun 20, 2022 at 15:33