An Invitation to Quantum Conditional Probability with Applications to Non-Invertible Symmetries and the Generalized Entropy

Abstract

Carrying the insights of conditional probability to the quantum realm is notoriously difficult due to the non-commutative nature of quantum observables. Nevertheless, conditional expectations on C* and 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 provide an overview of the multifaceted nature of quantum conditional probability by exploring different non-commutative generalizations of conditional expectations which meet the demands of non-exact error correction. We then exemplify the utility of these tools through two examples. First, we demonstrate how operator valued weights can be used to classify the possible symmetries (including non-invertible symmetries) of a quantum system. Then, we introduce a generalization of Connes’ spatial theory leading to a fully non-commutative form of 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 further provides an approach to factorizing the entropy of states into a sum of terms including an emergent area operator that is fully non-commutative rather than central.