Conventions and the explicit five-qubit encoding

All Hilbert spaces in the proved operational results are finite dimensional. Channels are completely positive and trace preserving (CPTP) on full matrix spaces. Set

The diamond norm includes arbitrary independent external references. Its standard finite-dimensional conventions are those of Watrous Watrous.

Let

Boundary legs are ordered $1,2,3,4,5$, and computational basis strings are ordered lexicographically with leg $1$ first. Addition in binary exponents is modulo two.

For $x=(x_1,\dots ,x_5)\in \{0,1{\}}^5$, define