Standard norm facts and existence of minimizers

For a difference of channels with input dimension $d$, the diamond norm can be evaluated on states with a reference of dimension $d$; a pure state suffices. CPTP pre- and postprocessing contract diamond distance, and channels have diamond norm one Watrous (Chapter 3). We use these standard norm facts explicitly.

The sets of density operators and finite-dimensional CPTP maps are compact. For channels this follows from their positive Choi matrices and the fixed partial-trace constraint. Hence both infima in Definitions def:decoder and def:leakage are attained.

For Hermitian operators, the trace-norm contraction used below also follows directly. If $X=X_{+}-X_{-}$ is its Jordan decomposition and $\Phi$ is CPTP, then

The same argument applies after adjoining a reference.