J equals five absorption-emission code

PDF p. 6: Æ-code bullet and Fig. 4

The article describes absorption–emission (Æ) codes as a family protecting against absorption–emission noise. Its selected $J=5$ instance uses the orthonormal magnetic basis $\mathrm{\mid }5,m⟩$, $m=-5,-4,\dots ,5$, and the two codewords $\mathrm{\mid }\bar{0}⟩=\sqrt{5\mathrm{/}7}\mathrm{\mid }5,-2⟩+\sqrt{2\mathrm{/}7}\mathrm{\mid }5,5⟩$ and $\mathrm{\mid }\bar{1}⟩=\sqrt{2\mathrm{/}7}\mathrm{\mid }5,-5⟩+\sqrt{5\mathrm{/}7}\mathrm{\mid }5,2⟩$. The code is their linear span. The encoding is the linear map from a two-dimensional logical space with orthonormal basis $\mathrm{\mid }0⟩,\mathrm{\mid }1⟩$ that sends these basis vectors to $\mathrm{\mid }\bar{0}⟩,\mathrm{\mid }\bar{1}⟩$, respectively.

For a precise error contract, choose a finite set $S$ of spin labels containing $5$, with $\mathrm{\mid }{J}^{\mathrm{\prime }}-5\mathrm{\mid }\le 1$ for every $J^{\mathrm{\prime }}\in S$. Let $H={⨁}_{J^{\mathrm{\prime }}\in S}{V}_{J^{\mathrm{\prime }}}$ be the orthogonal direct sum with one labelled copy of each block, let ${\iota }_{J^{\mathrm{\prime }}}:V_{J^{\mathrm{\prime }}}\to H$ be the inclusion, and let ${\pi }_5:H\to V_5$ be the orthogonal projection onto the input block. Regard the codewords and encoding as taking values in $H$ via ${\iota }_5$. For each $J^{\mathrm{\prime }}\in S$ and $k\in \{0,1\}$, take every component $T_q:V_5\to V_{J^{\mathrm{\prime }}}$, $-k\le q\le k$, of every rank-$k$ spherical tensor family in the error-span contract. The order-one absorption–emission error span is the complex linear span in $\mathcal{L}(H)$, the space of complex-linear endomorphisms of $H$, of $I_H$ and all ${\iota }_{J^{\mathrm{\prime }}}{T}_q{\pi }_5$. Rank refers to the tensor transformation law, not matrix rank. The article says this code corrects rank-one transitions with nonoverlapping error spaces. In the stated coherent error-span contract, the precise condition established below is $P{E}^{\ast }FP=c_{E,F}P$ for every two errors $E,F$ in the span, where $P$ is the orthogonal projector on $H$ onto the embedded code. In particular, $⟨\bar{0}\mathrm{\mid }{E}^{\ast }F\mathrm{\mid }\bar{1}⟩=0$: the error spaces generated from the two different codewords are orthogonal. This does not assert that the spaces generated by different errors on the same codeword are mutually orthogonal, or that every individually resolved matrix-unit transition belongs to the error span.

Angular-momentum basis and tensor labels

Code restriction and decoded channel types

Order-one absorption–emission tensor error span

The magnetic basis is orthonormal. Each displayed codeword has squared norm $5\mathrm{/}7+2\mathrm{/}7=1$, and their supports are disjoint, so they are orthogonal. Thus the encoding is an isometry and the code projector $P$ is well defined. Angular-momentum basis and tensor labels Code restriction and decoded channel types

The union of the magnetic supports is $\{-5,-2,2,5\}$, whose consecutive gaps are $3,4,3$. Both codewords have zeroth moment $1$, first moment $0$, and second moment $10$: for $\mathrm{\mid }\bar{0}⟩$ the latter two are $(5(-2)+2(5))\mathrm{/}7=0$ and $(5(4)+2(25))\mathrm{/}7=10$, and for $\mathrm{\mid }\bar{1}⟩$ they are $(2(-5)+5(2))\mathrm{/}7=0$ and $(2(25)+5(4))\mathrm{/}7=10$. The spaced-support moment criterion therefore gives the pair-product condition for the selected coherent tensor error span. Taking its matrix element between the orthogonal codewords gives the displayed cross-term vanishing and hence orthogonality of the two generated error spaces. The error-span contract defines the selected blocks, tensor components and pair-product condition; it does not identify the span with all resolved matrix units. Each ${\iota }_{J^{\mathrm{\prime }}}{T}_q{\pi }_5$ is an endomorphism of $H$, so the span and all products $P{E}^{\ast }FP$ are well typed. Order-one absorption–emission tensor error span Spaced-support moment criterion for order-one AE correction