Complete systems and operational definitions

preparations. If an ancillary state is mixed, choose a fixed purification and retain its purifying system in the complete dilation. A code-dependent initial environment is a different task: the code-to-total-state preparation must then itself be specified and tested for preservation.

coherences. With $E_{ij}=|i⟩⟨j|$, the complete channel is determined by

Agreement on $E_{ii}$ alone is not agreement on a quantum channel. For a qubit, both $E_{01}$ and $E_{10}={E_{01}}^{†}$ are required.