Notation, tasks, and elementary witnesses

All finite-system states and channels use ordinary matrix representations. A channel is a completely positive, trace-preserving (CPTP) linear map. For matrices and states, respectively, set

For channels with the same input and output types, define

For differences of finite-dimensional channels, this is the supremum of the output trace distance over referenced input states; a reference of input dimension suffices Watrous (Chapter 3).

Let $M$ be the input logical qubit, $Q$ an external reference, $R$ the accessible two-mode output, $E$ the residual two-mode output, and $M^{\prime }$ the decoded qubit. Logical bases of $M$ and $M^{\prime }$ are identified when comparing a decoded channel with ${\mathrm{id}}_M$. All entropy logarithms are base two.

Set

For a channel $\Lambda :M\to M^{\prime }$, define

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

These are target overlaps, hence squared fidelities against pure targets.

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

These are separate optimization problems. An optimal Bell-fidelity decoder need not minimize half-diamond error.

A replacement channel and a unitary channel are denoted by