Constructing resource theories of nonclassicality for continuous- and discrete-variable hybrid systems

Most of the physical systems used for quantum information processing consist of harmonic oscillators, described by continuous variables (CV), and effectively finite dimensional systems, described by discrete variables (DV). We propose a method of constructing a resource theory of nonclassicality for such a hybrid CV+DV system from that for a CV system. A monotone measure for the hybrid system can be obtained by performing an appropriate measurement on the DV system and then taking the average of the monotone measure for the conditional state of the CV system over measurement outcomes. As an example, we analyze the metrological power of a partially decohered Schrödinger's cat-like state. Our construction provides a general method of generalizing a convex resource theory defined on a subsystem to an extended system.