Conventions and elementary channel estimates

All Hilbert spaces are finite dimensional. We set $\hslash =1$. Entropy is in bits; logarithms in pulse-count estimates are natural. Write

For identified state and channel spaces, use

The diamond norm is the stabilized induced trace norm. For differences of channels with a $D$-dimensional input it may be evaluated on density operators with an untouched reference of dimension $D$. These distances contract under channel pre- and postcomposition Watrous (Chapter 3).

Define

for an isometry $V$ and a density operator $\tau$. For a channel $\mathcal{N}:C\to A$ and a specified complete complement $\mathcal{C}:C\to B$, define

where $\mathcal{D}$ is CPTP. Finite-dimensional compactness gives attainment of both infima.

The external reference is not available to the decoder. A half-diamond recovery bound controls every code–reference state, not merely a selected pure code input.