Æ rank-one correction with separated error spaces

PDF p. 6: Æ-code bullet continued into right column

Let $V_5$ have orthonormal magnetic basis $\mathrm{\mid }5,m⟩$, $m=-5,-4,\dots ,5$, and let the code be the span of $\mathrm{\mid }\bar{0}⟩=\sqrt{5\mathrm{/}7}\mathrm{\mid }5,-2⟩+\sqrt{2\mathrm{/}7}\mathrm{\mid }5,5⟩$ and $\mathrm{\mid }\bar{1}⟩=\sqrt{2\mathrm{/}7}\mathrm{\mid }5,-5⟩+\sqrt{5\mathrm{/}7}\mathrm{\mid }5,2⟩$. Select finitely many spin blocks including $V_5$, each with $\mathrm{\mid }{J}^{\mathrm{\prime }}-5\mathrm{\mid }\le 1$, and let $H$ be their orthogonal direct sum. Let $\mathcal{E}$ be the complex span on $H$ of the identity and the coherent components of rank-zero and rank-one spherical tensor families from $V_5$ into the selected blocks, extended by zero outside $V_5$. Tensor rank refers to the transformation law, not matrix rank.

For every $E,F\in \mathcal{E}$, $P{E}^{†}FP=c_{E,F}P$ for a scalar $c_{E,F}$, where $P$ is the code projector. Consequently every finite Kraus channel on $H$ whose Kraus operators lie in $\mathcal{E}$ has an exact decoded recovery on this code. In particular, $⟨\bar{0}\mathrm{\mid }{E}^{†}F\mathrm{\mid }\bar{1}⟩=0$, so the subspaces $\mathrm{span}\{E\mathrm{\mid }\bar{0}⟩:E\in \mathcal{E}\}$ and $\mathrm{span}\{F\mathrm{\mid }\bar{1}⟩:F\in \mathcal{E}\}$ are orthogonal. The article attributes this Æ code and its correction property to Ref. [9]. The conclusion concerns coherent tensor components; it does not claim correction of every separately resolved matrix-unit transition.