Complete interfaces and operational distances

A fixed mixed ancillary preparation is represented by a fixed purification whose unreturned purifying factor belongs to $E$. Input-dependent preparation must instead be included in the specified encoding map; it is not a fixed ancillary state.

For a state $\tau$, define the replacement channel

Our normalization is

where the diamond norm includes arbitrary untouched references. For differences of channels it is the supremum of the corresponding output trace norm on referenced input states Watrous (Chapter 3).