Populations and averaged realizations

Computational dephasing agrees with the identity on all computational populations but annihilates off-diagonal matrix units. It is not a quantum-code-preserving channel.

For a qubit, the two channels

have constant average $\mathrm{Tr}(\rho )I_2/2$, although each frozen realization perfectly distinguishes the computational basis inputs. Constancy of an averaged channel is not constancy of every realization.

If a realization-dependent certificate satisfies ${\mathbb{E}}_{\mathbf{J}}{e}_{\mathbf{J}}(t)\le a(t)$, Markov's inequality gives, with probability at least $1-p$,

Here $b_{\mathbf{J}}$ is a valid complete-channel comparison allowance for that realization. The decoder's dependence on the realization must be declared. A uniform time interval requires a separate uniform estimate.