J equals two detection encoding

PDF p. 6: Detection-code bullet and Fig. 4

In the spin-$2$ block $V_2={\mathbb{C}}^5$, let $\mathrm{\mid }2,m⟩$ for $m=-2,-1,0,1,2$ be the orthonormal magnetic basis of Angular-momentum basis and tensor labels. Define the detection encoding by the two codewords $\mathrm{\mid }\bar{0}⟩=(\mathrm{\mid }2,-2⟩+\mathrm{\mid }2,2⟩)\mathrm{/}\sqrt{2}$ and $\mathrm{\mid }\bar{1}⟩=\mathrm{\mid }2,0⟩$. Its code space is their complex linear span $C\subseteq V_2$. Equivalently, for a two-dimensional logical space with orthonormal basis $\mathrm{\mid }0⟩,\mathrm{\mid }1⟩$, its encoding is the linear map $V$ with $V\mathrm{\mid }0⟩=\mathrm{\mid }\bar{0}⟩$ and $V\mathrm{\mid }1⟩=\mathrm{\mid }\bar{1}⟩$; its code projector is $P=V{V}^{\ast }$ as in Code restriction and decoded channel types.

The article calls this an instance of the absorption–emission (Æ) codes of its Ref. [9], a family it describes as protecting against absorption–emission noise. Its Fig. 4 depicts the two nonzero $\mathrm{\mid }\bar{0}⟩$ amplitudes as $1\mathrm{/}\sqrt{2}$ and the nonzero $\mathrm{\mid }\bar{1}⟩$ amplitude as $1$. The name here specifies the encoding; which errors it detects or corrects is a separate conditional claim.

The five magnetic vectors are orthonormal, so the first displayed codeword has squared norm $(1+1)\mathrm{/}2=1$, the second has norm one, and their inner product is zero. Hence they form an orthonormal pair, $V$ is an isometry, and its image is exactly $C$. The code-isometry convention therefore makes $P=V{V}^{\ast }$ the orthogonal projector onto $C$. The family identification and figure description are attributed to the article and are not needed to establish these properties. Angular-momentum basis and tensor labels Code restriction and decoded channel types