Delocalized and Dynamical Catalytic Randomness and Information Flow

Abstract

We generalize the theory of catalytic quantum randomness to delocalized and dynamical settings. First, we expand the resource theory of randomness (RTR) by calculating the amount of entropy catalytically extractable from a correlated or dynamical randomness source. In doing so, we show that no entropy can be catalytically extracted when one cannot implement local projective measurement on randomness source without altering its state. The RTR, as an archetype of the `concave' resource theory, is complementary to the convex resource theories in which the amount of randomness required to erase the resource is a resource measure. As an application, we prove that quantum operation cannot be hidden in correlation between two parties without using randomness, which is the dynamical generalization of the no-hiding theorem. Second, we study the physical properties of information flow. Popularized quotes like "information is physical" or "it from bit" suggest the matter-like picture of information that can travel with the definite direction while leaving detectable traces on its region of departure. To examine the validity of this picture, we focus on that catalysis of randomness models directional flow of information with the distinguished source and recipient. We show that classical information can always spread from its source without altering its source or its surrounding context, like an immaterial entity, while quantum information cannot. We suggest an approach to formal definition of semantic quantum information and claim that utilizing semantic information is equivalent to using a partially depleted information source. By doing so, we unify the utilization of semantic and non-semantic quantum information and conclude that one can always extract more information from an incompletely depleted classical randomness source, but it is not possible for quantum randomness sources.