Von Neumann Algebras are a very sophisticated topic in modern day functional analysis and have come into focus ever since the Connes Embedding Conjecture was resolved using Complexity Theory. These notes however are meant to stress the case for the generalised notion of entropy for a quantum state on a Von Neumann Algebra that includes classical and quantum parts.
Much of the discussion that is presented is taken directly from a paper of Harlow on "The Ryu-Takayanagi Formula from Quantum Error Correction" - I do not claim any originality in what is presented (except for the beautiful image above) - the Harlow style presentation I feel is best suited for physicists who need only to grasp the basics of this complicated subject.
