Lemma: Environmental distinguishability from a Bell test

Let $\mathcal{N}$ be a qubit-input channel with a fixed complete complement. If a decoder has Bell fidelity $F_e$, then the complementary outputs for any orthogonal qubit pair have trace distance at most

There is also a replacement state $\tau$ on the original complete environment with

Proof. Dilate the decoder and apply the complete evolution to a Bell input. The result is a pure vector $|\Psi ⟩$ on reference, decoded qubit, and enlarged environment. Its projection onto the target Bell vector has norm $\sqrt{F_e}$. For $F_e>0$, normalize that projection to an environmental vector $|e⟩$. Then

The assertion for $F_e=0$ is trivial. Tracing the decoded qubit and decoder environment shows that the normalized Choi state of the original complement is within $\sqrt{1-F_e}$ of $I_R/2\otimes \tau$, for the corresponding reduced $\tau$.

For an orthogonal input pair with Bloch direction $L$,

The complete trace norm of the difference bounds this partial contraction. Hence

which is the first assertion.

For a qubit-input Hermiticity-preserving map $\Lambda$,

Indeed every pure referenced qubit input is $(A\otimes I)|\Phi ⟩$, with $\Vert A{\Vert }^2\le 2$; apply the trace-norm product bound and then convexity. Reference dimension two suffices by Schmidt decomposition. The Choi estimate proves the replacement-channel bound.