Complete channels, access, and distances

Let $C\simeq {\mathbb{C}}^d$, with $d\ge 2$, be the logical code. All output Hilbert spaces are separable. States are normal density operators, and channels are normal completely positive trace-preserving maps on trace-class operators. We abbreviate “completely positive and trace preserving” as CPTP.

We use

The diamond norm is the reference-stable trace-norm distance. For a finite input of dimension $d$, a reference of dimension $d$ suffices for channel discrimination; the usual finite-input conventions are as in Watrous.

For an isometry

define

The inaccessible output $B$ is complete: it includes uncollected modes, retained source degrees of freedom, displaced carriers, preparation purifiers, and inaccessible apparatus factors. A factor is not removed from this ledger merely because an observer does not control it.

For a state $\sigma$, the replacement channel is

Set

The infimum in (eq:erec) is over deterministic decoders. A heralded success branch, by itself, is not such a decoder.