Informal Physics Half-Hour Talk

Abstract: Conditional expectations on von Neumann algebras have played a significant role in the development of quantum information theory, and especially the study of quantum error correction. In quantum gravity, it has been suggested that conditional expectations may be used to implement the holographic map algebraically, with quantum error correction underlying the emergence of spacetime through the generalized entropy formula. However, the requirements for exact error correction are almost certainly too strong for realistic theories of quantum gravity. In this talk, we present a relaxed notion of quantum conditional expectation which implements non-exact error correction. We introduce a generalization of Connes’ spatial theory adapted to completely positive maps, and derive a chain rule allowing for the non-commutative factorization of relative modular operators into a marginal and conditional part, resembling Bayes’ law. This allows for an exact quantification of the information gap occurring in the data processing inequality for arbitrary quantum channels. When applied to algebraic inclusions, this also provides an approach to factorizing the entropy of states into a sum of terms which may be interpreted as a generalized entropy. We illustrate that the emergent area operator is fully non-commutative rather than central, except under the conditions of exact error correction. We argue that this may be interpreted as a consequence of quantum corrections brought about by the backreaction of geometry and matter.