Complete channels, access, and local recovery facts

Let $C\simeq {\mathbb{C}}^d$, $d\ge 2$, be the logical code. Output Hilbert spaces are separable. States are normal density operators, and channels are normal completely positive trace-preserving maps on trace-class operators, abbreviated CPTP. The finite programs below use finite-dimensional encodings.

Define

The diamond distance retains arbitrary external references. For finite input dimension $d$, a reference of dimension $d$ suffices in channel discrimination Watrous.

For an isometry

write

The output $R$ is declared accessible. The residual $B$ is complete relative to that declaration: it contains every inaccessible output of the specified pure dilation, including preparation purifiers and apparatus factors when present.

For a state $\sigma$, define

The two diagnostics are

Decoders are deterministic. A normalized postselected branch is not, by itself, a candidate in (eq:rec).

For a fixed code basis, put

and define the normalized Choi state

Agreement on the populations $E_{ii}$ alone is not agreement on the quantum code.