I will go over the new proof of the quantum null energy condition (QNEC):
The proof uses tools and techniques from the modular theory. In particular, see Section 2.4 theorem B of Borchers notes:
An important concept for the proof is the so-called relative modular flow.
For a definition of the relative modular operator see Edward's notes section 3.2:
An important property of the relative modular flow for us is V.2.41 and V.2.42 in Section V.2.3 of Haag's book "Local Quantum Physics".
You can find more information about it in Section 2.4 of https://arxiv.org/pdf/1811.05052.pdf .