Finite interfaces and normalization

All Hilbert spaces in Sections sec:interfaces–sec:stability are finite dimensional. A continuum or gravitational extension requires additional hypotheses and is not implicit.

For a Hilbert space ${\mathcal{H}}_X$, write $\mathcal{B}({\mathcal{H}}_X)$ for its matrix algebra and $\mathsf{S}({\mathcal{H}}_X)$ for its density operators. Channels are linear, completely positive, trace-preserving maps. For a channel difference, the diamond norm is stabilized against an arbitrary untouched reference. We use the conventions

These distances are at most one for normalized states, channels, and probability laws. Standard channel facts used here can be found in Watrous.

An external reference is denoted by $Q$. It is not an output environment and is not traced out in a reference-preservation claim.