Lemma: Triple channel and logical representatives

For $\rho =(I+r_{x}X+r_{y}Y+r_{z}Z)/2$,

The representatives satisfy

Proof. For $R_Z$, multiplication by $\overline{Z}$ gives $R_Z\overline{Z}=g_3$, so its action on the code is logical $Z$. For $R_X$, $R_X\overline{X}=YIYXX$. The latter is the cyclic shift by two positions of $g_2{g}_3$, and is a stabilizer. Cyclic shifts preserve the code because (eq:Vmatrix) is cyclically invariant. Thus $R_X$ acts as logical $X$. Direct Pauli multiplication then gives the $R_Y$ assertions.

For completeness, the syndrome map on the six binary Pauli coordinates of $123$ has rank four. For example the syndromes of $X_1,X_2,Z_2,Z_3$ are independent. Its kernel has dimension two, and its three nonidentity elements are the unsigned versions of $R_X,R_Y,R_Z$. Every other nonidentity Pauli on the triple compresses to zero. Also ${\mathcal{N}}_{123}(I/2)=I_8/8$ by Proposition prop:perfect. Pauli expansion now gives (eq:N123).