Supermaps between channels of any type

Abstract

Supermaps between quantum channels (completely positive trace-preserving (CPTP) maps of matrix algebras) were introduced in [Chiribella et al., EPL 83(3) (2008)]. In this work we generalise to supermaps between channels of any type; by channels we mean CPTP maps of finite-dimensional C*-algebras. Channels include POVMs, quantum instruments, classically controlled families of quantum channels, classical channels, quantum multimeters, and more. We show that deterministic supermaps between channels of any type can be realised using simple circuits, recovering the previous realisation theorems of [Chiribella et al., EPL 83(3) (2008)] (for deterministic supermaps between quantum channels) and [Bluhm et al. (2024)] (for deterministic supermaps between quantum multimeters) as special cases. To prove this realisation theorem we use the graphical calculus of the 2-category of finite-dimensional 2-Hilbert spaces; the paper includes an accessible and elementary introduction to this graphical calculus, and no prior knowledge of category theory is expected of the reader.