Random Pauli measurement updates stabilizer generators

Each unchanged row commutes with $M$. If $g_j$ anticommutes with $M$, then $g_j{g}_1$ commutes with $M$ because both factors anticommute with $M$. The changed and unchanged rows commute with each other because they are products of elements of the old abelian group. The new pivot $(-1{)}^{b}M$ commutes with each other row by the first two calculations and is Hermitian. Every changed row is Hermitian because its two factors commute and are Hermitian; its exact phase-aware product is the one computed by [A1].

Write $v_j=\lambda (g_j)$ and $m=\lambda (M)$. By [F6] and [F8], the updated rows for $j>1$ have labels $v_j^{\mathrm{\prime }}=v_j+\omega (v_j,m)v_1$. They remain independent: in any zero linear combination, the coefficients of the independent vectors $v_2,\dots ,v_n$ are the coefficients of the updated rows, so each is zero. Also $\omega (v_j^{\mathrm{\prime }},m)=\omega (v_j,m)+\omega (v_j,m)\omega (v_1,m)=0$; unchanged rows already pair to zero with $m$. Every $v_j^{\mathrm{\prime }}$ lies in $L$, but $m\mathrm{\notin }L$ because it pairs nontrivially with $v_1$ while isotropy in [F5] makes every vector in $L$ pair trivially with $v_1$. Thus adjoining $m$ gives $n$ independent labels. Any nonempty product of the new commuting generators has one of these nonzero labels, so it cannot equal the zero-label scalar $-I$. The generated subgroup therefore excludes $-I$.

Since $g_1$ anticommutes with $M$, [F1] gives probability $1\mathrm{/}2$ to each bit; [F4] therefore gives $\mathrm{\parallel }{P}_b\mathrm{\mid }\psi ⟩{\mathrm{\parallel }}^2=1\mathrm{/}2$, so the branch vector is nonzero. The new pivot stabilizes it because $M{P}_b=(-1{)}^b{P}_b$. Each unchanged row commutes with $P_b$ and fixes $\mathrm{\mid }\psi ⟩$. For an updated row $g_j{g}_1$, commutation with $P_b$ and $g_j{g}_1\mathrm{\mid }\psi ⟩=\mathrm{\mid }\psi ⟩$ give $g_j{g}_1{P}_b\mathrm{\mid }\psi ⟩=P_b\mathrm{\mid }\psi ⟩$. Thus all updated generators fix the normalized branch.

The new generators have rank $n$, so their common fixed space is one-dimensional by Stabilizer projector rank and code dimension. It contains the normalized post-measurement vector from step 2.2, hence it is exactly that post-measurement ray.