Classical–quantum cuts

A cut with finite public history set $H$ and branch spaces ${\mathcal{H}}_h$ has algebra

A state is a family of positive blocks $({\rho }_h{)}_h$ with ${\sum }_h\mathrm{Tr}{\rho }_h=1$. Equivalently, it is the block-diagonal operator

The quantum names in ${\mathcal{H}}_h$, their locations, and their ownership are part of the cut specification. Histories may have different live spaces. Direct sums keep those types distinct.

Whenever a full matrix extension is needed, the specified extension first pinches between the classical history sectors. It does not interpret a coherent superposition of controller histories as an ordinary classical control.

An untouched reference $E$ has no execution or communication wire. All channel identities below hold after tensoring with ${\mathrm{id}}_E$.