Minimal single-manifold encoding

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

Use the spin-$1$ block $V_1={\mathbb{C}}^3$ with its orthonormal magnetic basis $\mathrm{\mid }1,m⟩$, $m=-1,0,1$. The minimal single-manifold encoding has logical codewords $\mathrm{\mid }\bar{0}⟩=\mathrm{\mid }1,-1⟩$ and $\mathrm{\mid }\bar{1}⟩=\mathrm{\mid }1,1⟩$. Its codespace is their complex linear span $\mathcal{C}\subset V_1$. Equivalently, its encoding is the linear map $V:{\mathbb{C}}^2\to V_1$ sending the standard logical basis vectors $\mathrm{\mid }0⟩,\mathrm{\mid }1⟩$ to $\mathrm{\mid }\bar{0}⟩,\mathrm{\mid }\bar{1}⟩$, respectively.

Angular-momentum basis and tensor labels

Finite operator and ket conventions

Code restriction and decoded channel types

The angular-basis convention supplies the three distinct orthonormal vectors in $V_1$; the finite operator conventions give the ket notation. Thus the two displayed codewords are orthonormal, and their linear span is a two-dimensional subspace. For logical vectors $x=x_0\mathrm{\mid }0⟩+x_1\mathrm{\mid }1⟩$ and $y=y_0\mathrm{\mid }0⟩+y_1\mathrm{\mid }1⟩$, orthonormality gives the same inner product for $Vx,Vy$ as for $x,y$. Hence $V$ is an isometry and defines a code in the stated code-restriction convention. Angular-momentum basis and tensor labels Finite operator and ket conventions Code restriction and decoded channel types

The article calls this the smallest encoding supported in a single $J$ manifold and uses it as its lowest-$J$ baseline for detecting a decay: in its stated model a decay is flagged because the state leaves the $J=1$ manifold. Its Fig. 4 depicts unit amplitudes at $m=-1,+1$. These are descriptions of the article's benchmark; the definition alone does not assert a universal minimality theorem or specify a decay channel.