Theorem: Exact response and recovery for the deformation

For (eq:rotationisometry),

and

The exact quantities are

For $0<\theta <\pi /2$, the unitary $U_{\theta }$ is non-Clifford and its output on the stabilizer input $|+⟩|0⟩$ has positive bipartite non-local magic across $R:B$. At $\theta =\pi /2$, the unitary is Clifford but the complement is still nonconstant.

Proof. Partial traces of the two columns in (eq:rotationisometry) give the displayed channels. The residual pure states have overlap $\cos \theta$, hence trace distance $\sin \theta$. Lemma lem:witness gives the lower bound ${\delta }_{\mathrm{full}}\ge \sin \theta /2$.

Let

Each basis output has distance $\sin \theta /2$ from ${\sigma }_{\theta }$. For an arbitrary referenced input, the output difference is

where the diagonal reference blocks $X_i$ are positive and ${\sum }_i\mathrm{Tr}{X}_i=1$. The trace norm is therefore at most $\sin \theta$. This proves the matching half-diamond upper bound.

The radiation channel has

The identity decoder has error $(1-c)/2$, using an upper triangle bound and the orthogonal Bell outputs of $I$ and $Z$. For any decoder of error $\epsilon$, the ideal outputs of $|+⟩$ and $|-⟩$ have distance one, while their channel outputs have distance $c$. Contraction and the triangle inequality give $1\le 2\epsilon +c$. This proves the recovery formula.

For input $|+⟩$, the complete output is

Its radiation reduction has eigenvalues $(1+\cos \theta )/2$ and $(1-\cos \theta )/2$. For $0<\theta <\pi /2$, these are nonzero and unequal. Lemma lem:flat shows that the state is not locally equivalent to a stabilizer state. In particular, $U_{\theta }$ cannot be Clifford, because it maps a stabilizer input to a nonstabilizer output.

At $\theta =\pi /2$, $R_y(\pi )=XZ$, and

where $\mathrm{CZ}$ acts first. This is Clifford. On the allocated $|0⟩$ residual, its code isometry is the copying isometry of Proposition prop:independence.