(1) The ideas of entanglement and quantum information theory have invaded both quantum field theory and string theory and the developments are so extensive that even a long document of 70 pages is short to mention thes relevant issues.

So Witten refers us to the following review. This itself is 60 pages long.

[1] T. Nishioka, “Entanglement Entropy: Holography and Renormalization Group,” arXiv:1801.10352.

(2) The object in his notes is to collect the mathematical ideas that are relevant at a place and not to offer a review of the topic. The ideas are not easily available at a single place and hence we have to be thankful to him.

(3) He makes contact with older ideas of axiomatic and algebraic QFT. This is a topic that is obtuse because of its complexity as well as mathematical rigor. I was amongst those who used to think that we shall not need this kind of technology in future. Apparently I was wrong.

(4) Then we come to the Reech-Schlieder Theorem. It says that adjescent regions of QFT spacetime are entangled and the entanglement is ultraviolet divergent. This is true even without quantum gravity.

This is where the rub will lie. We though that entanglement is some quirk of black hole physics. We already have inconvenient things in QFT.

(5) The entanglement is not merely a property of the states but the algebras of observables itself.

(6) Witten's article aims at explaining these issues.

(7) Tomita-Takesaki Theory is a tool that helps us in those cases when entanglement is a property of the algebra of observables itself rather than the states.

(8) T3 is applicable in so many cases that Witten decides not to record all the references but describes an example in section 2.

(9) H. Araki's work too comes into focus in this regard.

(10) Then there is this talk about peeping behind the black hole horizon.

(11) Then we start seeing those typical terms that we see only in complex analysis - like monotonicity.

(12) That ends the preview of Section 2.

So Witten refers us to the following review. This itself is 60 pages long.

[1] T. Nishioka, “Entanglement Entropy: Holography and Renormalization Group,” arXiv:1801.10352.

(2) The object in his notes is to collect the mathematical ideas that are relevant at a place and not to offer a review of the topic. The ideas are not easily available at a single place and hence we have to be thankful to him.

(3) He makes contact with older ideas of axiomatic and algebraic QFT. This is a topic that is obtuse because of its complexity as well as mathematical rigor. I was amongst those who used to think that we shall not need this kind of technology in future. Apparently I was wrong.

(4) Then we come to the Reech-Schlieder Theorem. It says that adjescent regions of QFT spacetime are entangled and the entanglement is ultraviolet divergent. This is true even without quantum gravity.

This is where the rub will lie. We though that entanglement is some quirk of black hole physics. We already have inconvenient things in QFT.

(5) The entanglement is not merely a property of the states but the algebras of observables itself.

(6) Witten's article aims at explaining these issues.

(7) Tomita-Takesaki Theory is a tool that helps us in those cases when entanglement is a property of the algebra of observables itself rather than the states.

(8) T3 is applicable in so many cases that Witten decides not to record all the references but describes an example in section 2.

(9) H. Araki's work too comes into focus in this regard.

(10) Then there is this talk about peeping behind the black hole horizon.

(11) Then we start seeing those typical terms that we see only in complex analysis - like monotonicity.

(12) That ends the preview of Section 2.