Conventions and common-space comparison

Set $\hslash =1$. The code ${\mathcal{H}}_C={\mathbb{C}}^d$, $d\ge 2$, is finite dimensional. Output Hilbert spaces may be infinite dimensional and separable. Channels are normal CPTP maps on trace-class operators. We use

An untouched reference of dimension $d$ suffices for discrimination of channels with this finite input. For the complete isometric channels below, the span of the two output ranges is finite dimensional, so the corresponding comparison reduces directly to finite-dimensional channel discrimination Watrous (Chapter 3).

Write

For an accessible channel $\mathcal{N}:C\to A$ and specified complete complement $\mathcal{C}:C\to B$, define

Explicit optimizers will be constructed. No general infinite-dimensional compactness claim is needed.