Notation and referenced channel metrics

For a finite matrix $X$, write

For states,

For channels with the same input and output spaces,

For a difference of finite-dimensional channels, this equals the supremum of the output trace distance over referenced input states. A reference of input dimension suffices. These are standard finite-channel conventions Watrous.

Let $M$ be a logical qubit, $Q$ an external reference, $R$ the accessible two-mode output, and $E$ the residual two-mode output. The decoded qubit is $M^{\prime }$. All logarithms in entropy and entanglement expressions are base two.

For a qubit-output channel $\Lambda$, define

For a purification $|{\psi }_{\rho }{⟩}_{QM}$ of ${\rho }_M$, define

The target overlaps are squared fidelities against their pure targets.

For an accessible channel $\mathcal{N}:M\to R$, set

where the input and decoded logical bases are identified.

A replacement channel is

and ${\mathrm{Ad}}_U(X)=UX{U}^{†}$.