doc: add section on sector changes
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@ -584,6 +584,194 @@ The $q=0$ value $\tfrac13$ is $\varphi$-independent, because $ZZZZ$ is diagonal
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This example illustrates Corollary~\ref{cor:joint-refinement} concretely: the full $9\times9$ block factors, without any state-specific input, into one $3\times3$ block of rank $1$, two mutually zero $2\times2$ blocks, and two $1\times1$ blocks, purely by charge conservation --- before any computation is needed to know which entries can possibly be nonzero. Since $\rho_\varphi$ is generically not a stabilizer state, this degeneracy pattern cannot be attributed to Lemma~\ref{lem:stabilizer-degeneracy}, giving a concrete instance of the separation of mechanisms described in general terms by Remark~\ref{rem:two-mechanisms}.
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\subsection{Full collective $SU(2)$ symmetry: branching, exact projectors, and cut-independent data}
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\label{sec:full-su2-symmetry}
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% ============================================================
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% NEW SUBSECTION -- draft, inserted between the U(1) worked example
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% (sec:u1-example) and the stabilizer mechanism. This makes good on
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% the remark, already present in rem:abelian-selection-rule, that the
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% non-abelian case needs the Casimir rather than a Cartan generator.
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% Structure follows the session plan: (1) branching formula,
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% (2) projector formula + exact Casimir construction, (3) worked
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% example on a 6-qubit singlet-network state, (4) cut-independent
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% ("recoupling") data via a coherence-term decomposition.
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% Numerics were produced and cross-checked (brute-force simulation
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% AND independent Casimir-diagonalization route agree to machine
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% precision) with the accompanying scripts; see the TODO comments
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% below for what is NOT yet nailed down.
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% ============================================================
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Remark~\ref{rem:abelian-selection-rule} already anticipated that the non-abelian case needs the Casimir operator, not a Cartan generator, and stops short of working this out. We do so here, for the strong hypothesis of Section~\ref{sec:mechanisms} specialized further: not merely party-local, but \emph{collective}, i.e.\ the same single-qubit unitary $U_g$ acting on every party of the full system $P$ simultaneously,
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\begin{equation}
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g\ \longmapsto\ \bigotimes_{a\in P}U_g,
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\qquad
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g\in SU(2).
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\label{eq:collective-hypothesis}
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\end{equation}
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Since $\mathcal V_0^{(a)}\cong\mathbb R^3$ carries the adjoint representation $\mathrm{Ad}_{U_g}$, which factors through $SO(3)=SU(2)/\{\pm\id\}$, only integer angular momenta $j$ occur throughout this subsection; we write $G=SO(3)$ accordingly. Eq.~\eqref{eq:collective-hypothesis} is a special case of the strong, party-local hypothesis of Section~\ref{sec:mechanisms} (hence also of the weak, cut-factorizing one), so Proposition~\ref{prop:block-diagonal} and Proposition~\ref{prop:sector-symmetry-compatible} both apply to every cut simultaneously, with the \emph{same} group element $g$ on both sides of every cut -- this simultaneity is what Section~\ref{sec:cut-independent-data} below exploits.
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\subsubsection{Branching}
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For a cluster of $k$ qubits, $\mathcal V_0^{(V)}\cong(\mathbb R^3)^{\otimes k}$ decomposes under $G$ as $\bigoplus_j m_j^{(k)}V_j$. The multiplicities are already implicit in Remark~\ref{rem:abelian-selection-rule}'s weight count $N_M^{(k)}$, now taken for general $k$ rather than only $k=2$.
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\begin{proposition}[Branching multiplicities from $U(1)$ weight counts]
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\label{prop:branching-su2}
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Let $N_M^{(k)}$ be the dimension of the weight-$M$ eigenspace of the collective Cartan generator $J_z^{\mathrm{tot}}$ on $(\mathbb R^3)^{\otimes k}$, i.e.\ the coefficient of $x^M$ in $(x^{-1}+1+x)^k$ (Remark~\ref{rem:abelian-selection-rule}, generalized from $k=2$). Then
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\begin{equation}
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m_j^{(k)} = N_j^{(k)} - N_{j+1}^{(k)},
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\qquad j=0,1,\dots,k,
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\label{eq:branching-formula}
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\end{equation}
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with $N_{k+1}^{(k)}:=0$.
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\end{proposition}
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\begin{proof}
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Each irreducible $V_j$ contributes exactly one basis vector of weight $M$ for every integer $M$ with $\lvert M\rvert\le j$, and none otherwise. Summing over the isotypic decomposition, $N_M^{(k)}=\sum_{j\ge\lvert M\rvert}m_j^{(k)}$ for every $M\ge0$; the claim follows by taking the difference of consecutive $M$.
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\end{proof}
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For $k=3$ (the source cluster size used throughout the worked example below), the trinomial coefficients of $(x^{-1}+1+x)^3$ give $N_0^{(3)}=7$, $N_1^{(3)}=6$, $N_2^{(3)}=3$, $N_3^{(3)}=1$, hence
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\begin{equation*}
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m_0^{(3)}=1,\qquad m_1^{(3)}=3,\qquad m_2^{(3)}=2,\qquad m_3^{(3)}=1,
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\end{equation*}
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matching the dimension count $\sum_j(2j+1)m_j^{(3)}=1+9+10+7=27=3^3$. This is the first case with a nontrivial multiplicity ($m_1^{(3)}=3$), which is why the worked example below is built on three-qubit clusters.
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\subsubsection{The reduced block, without Clebsch--Gordan coefficients}
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\begin{corollary}[Isotypic norm without an explicit coupled basis]
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\label{cor:projector-norm-formula}
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Under the collective hypothesis~\eqref{eq:collective-hypothesis}, fix a cut $S\mid S^c$ and sectors $V\subseteq S$, $T\subseteq S^c$, and let $P_j^{(V)}$, $P_j^{(T)}$ denote the orthogonal projectors onto the spin-$j$ isotypic components of $\mathcal V_V^{(S)}\cong(\mathbb R^3)^{\otimes\lvert V\rvert}$, $\mathcal V_T^{(S^c)}\cong(\mathbb R^3)^{\otimes\lvert T\rvert}$. Then, writing $M:=M_{V\to T}(\rho)$,
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\begin{equation}
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M = \bigoplus_j P_j^{(T)}\,M\,P_j^{(V)},
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\qquad
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\|A_j\|_* = \frac{1}{2j+1}\bigl\|P_j^{(T)}\,M\,P_j^{(V)}\bigr\|_*,
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\qquad
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\|M\|_* = \sum_j (2j+1)\,\|A_j\|_*,
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\label{eq:projector-norm-formula}
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\end{equation}
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with all three statements exact, not merely bounds.
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\end{corollary}
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\begin{proof}
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By Proposition~\ref{prop:block-diagonal}, $M$ is already block diagonal across isotypes with $M|_j=P_j^{(T)}MP_j^{(V)}=A_j\otimes\id_{2j+1}$ and $M|_j P_{j'}^{(V)}=0$ for $j'\ne j$; since the $P_j^{(V)}$ (resp.\ $P_j^{(T)}$) are mutually orthogonal projectors summing to the identity, $M=\bigoplus_jM|_j$ is an honest orthogonal direct sum on \emph{both} sides, and the singular values of a matrix with orthogonally supported blocks are exactly the union of the blocks' singular values. The nuclear norm of $A_j\otimes\id_{2j+1}$ is $(2j+1)\|A_j\|_*$, since each singular value of $A_j$ is repeated $2j+1$ times.
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\end{proof}
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Eq.~\eqref{eq:projector-norm-formula} means $\|A_j\|_*$ -- and hence the sector-resolved witness $(2j+1)\|A_j\|_*>1$ of Corollary~\ref{cor:sub-block} applied to $\Pi_{\mathcal V}=P_j^{(V)}$, $\Pi_{\mathcal T}=P_j^{(T)}$ -- is computable from the isotypic projectors \emph{alone}, without ever constructing the coupled (Clebsch--Gordan) basis $\ket{j,m;\mu}$ that Eq.~\eqref{eq:wigner-eckart} presupposes. Only the finer question of \emph{which direction} in the multiplicity space $M_j^{(V)}$ carries the signal -- the "diagnostic resolution" motivation of Section~\ref{sec:introduction} -- needs the coupled basis and hence genuine Clebsch--Gordan data.
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\begin{remark}[Exact projectors via the Casimir, not group averaging]
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\label{rem:casimir-projectors}
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The projectors $P_j$ in Corollary~\ref{cor:projector-norm-formula} can be obtained without any integration over $G$. On $(\mathbb R^3)^{\otimes k}$, let $J_a^{\mathrm{tot}}:=\sum_{l=1}^k\id^{\otimes(l-1)}\otimes J_a\otimes\id^{\otimes(k-l)}$ for $a\in\{x,y,z\}$, where $J_a$ is the spin-$1$ generator $(J_a)_{bc}=-i\epsilon_{abc}$ on a single leg, and set $J^2_{\mathrm{tot}}:=\sum_aJ_a^{\mathrm{tot}}J_a^{\mathrm{tot}}$. Since $J^2_{\mathrm{tot}}$ is Hermitian and its eigenvalues on the isotypic component $V_j$ are exactly $j(j+1)$, the eigenspace projectors of $J^2_{\mathrm{tot}}$ \emph{are} the $P_j$, obtainable by a single Hermitian eigendecomposition of a $3^k\times3^k$ matrix -- exact up to ordinary floating-point roundoff, with no Monte-Carlo integration over $SO(3)$ required. This is the construction used in \texttt{scripts/exact\_casimir\_projectors.py}; an earlier Monte-Carlo character-averaging version (integrating $\chi_j(g)\,D(g)$ against Haar measure) was used during development and is superseded by this exact construction throughout.
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\end{remark}
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\subsubsection{Worked example: a six-qubit singlet network}
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\label{sec:singlet-network-example}
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Fix six qubits $A,B,C,D,E,F$ and the cut $S=\{A,B,C\}\mid S^c=\{D,E,F\}$, together with the finer cut $S'=\{A,B\}\mid S'^c=\{C,D,E,F\}$. Let
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\begin{equation}
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\ket{\psi_1}:=\ket{\psi^-}_{AD}\otimes\ket{\psi^-}_{BE}\otimes\ket{\psi^-}_{CF},
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\qquad
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\ket{\psi_2}:=\ket{\psi^-}_{AE}\otimes\ket{\psi^-}_{BF}\otimes\ket{\psi^-}_{CD},
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\label{eq:singlet-network-states}
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\end{equation}
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with $\ket{\psi^-}=(\ket{01}-\ket{10})/\sqrt2$, two different perfect matchings of the six qubits into singlets, each straddling both cuts. Both states are pure singlets of the \emph{full} collective rotation $U_g^{\otimes6}$ -- each factor is individually annihilated by $\sum_aJ_a^{\mathrm{tot}}$ -- so every state in $\mathrm{span}\{\ket{\psi_1},\ket{\psi_2}\}$ satisfies the collective hypothesis~\eqref{eq:collective-hypothesis} for the \emph{same} $G$, on \emph{both} cuts simultaneously.
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\begin{example}[Aligned singlets: an exactly solvable check]
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\label{ex:aligned-singlets}
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For $\rho_1=\ketbra{\psi_1}{\psi_1}$, the standard singlet correlator $\bra{\psi^-}\sigma_i\otimes\sigma_j\ket{\psi^-}=-\delta_{ij}$ and independence of the three pairs give, for all $i_A,\dots,i_F\in\{x,y,z\}$,
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\begin{equation}
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\tr\bigl(\rho_1\,\sigma_{i_A}\!\otimes\sigma_{i_B}\!\otimes\sigma_{i_C}\!\otimes\sigma_{i_D}\!\otimes\sigma_{i_E}\!\otimes\sigma_{i_F}\bigr)
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= -\,\delta_{i_Ai_D}\delta_{i_Bi_E}\delta_{i_Ci_F}.
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\label{eq:aligned-singlet-tensor}
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\end{equation}
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At the cut $S\mid S^c$, reshaping Eq.~\eqref{eq:aligned-singlet-tensor} into the $27\times27$ matrix $M_{S\to S^c}(\rho_1)$ (rows $(i_A,i_B,i_C)$, columns $(i_D,i_E,i_F)$) gives exactly $M_{S\to S^c}(\rho_1)=-\id_{27}$. A scalar multiple of the identity is scalar in \emph{every} orthonormal basis, so Corollary~\ref{cor:projector-norm-formula} gives $A_j=-\id_{m_j^{(3)}}$ with no computation beyond Proposition~\ref{prop:branching-su2}: $\|A_j\|_*=m_j^{(3)}$, i.e.\ $1,3,2,1$ for $j=0,1,2,3$, and $\sum_j(2j+1)\|A_j\|_*=27=\|M_{S\to S^c}(\rho_1)\|_*$.
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At the finer cut $S'\mid S'^c$, the same tensor~\eqref{eq:aligned-singlet-tensor}, reshaped to $9\times81$, has exactly $9$ nonzero entries per row (fixed $i_A,i_B$: the entries $(i_C,i_D,i_E,i_F)=(t,i_A,i_B,t)$ for $t\in\{x,y,z\}$), with pairwise disjoint column supports across rows, so $M_{S'\to S'^c}(\rho_1)$ has $9$ equal singular values $\sqrt3$. Here $M_{S'\to S'^c}(\rho_1)M_{S'\to S'^c}(\rho_1)^T=3\id_9$ exactly, so \emph{every} orthonormal sub-basis of the domain -- in particular each isotypic sector $j=0,1,2$ of the two-qubit source $\{A,B\}$ -- sees the same image norm $\sqrt3$; the resulting equality $\|A_0\|_*\approx\|A_1\|_*\approx\|A_2\|_*\approx\sqrt3$ is therefore a linear-algebra consequence of this isometry property rather than a nontrivial fact about $SO(3)$.
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\end{example}
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\begin{example}[A superposition: genuine multiplicity structure and a coherence term]
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\label{ex:singlet-superposition}
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Let $\ket{\Xi(\alpha)}:=(\cos\alpha\ket{\psi_1}+\sin\alpha\ket{\psi_2})/N(\alpha)$, still collectively invariant for every $\alpha$ since $\mathrm{span}\{\ket{\psi_1},\ket{\psi_2}\}$ is a $G$-fixed subspace. Because $\rho_\Xi=\ketbra{\Xi}{\Xi}$ is \emph{not} an affine combination of $\rho_1,\rho_2$ for a coherent (rather than incoherent) superposition, the correlation tensor of $\rho_\Xi$ picks up a coherence term absent from Example~\ref{ex:aligned-singlets}:
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\begin{equation}
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T(\Xi(\alpha)) = \frac{\cos^2\!\alpha\;T_1 + \sin^2\!\alpha\;T_2 + \cos\alpha\sin\alpha\;C_{12}}{N(\alpha)^2},
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\qquad
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N(\alpha)^2=1+\sin(2\alpha)\,\mathrm{Re}\!\braket{\psi_1}{\psi_2},
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\label{eq:coherence-decomposition}
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\end{equation}
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where $T_a(\vec\imath):=\bra{\psi_a}\sigma_{\vec\imath}\ket{\psi_a}$ ($a=1,2$; $T_1$ is Eq.~\eqref{eq:aligned-singlet-tensor}) and $C_{12}(\vec\imath):=\bra{\psi_1}\sigma_{\vec\imath}\ket{\psi_2}+\bra{\psi_2}\sigma_{\vec\imath}\ket{\psi_1}$ is a third, fixed (state-family-, not $\alpha$-dependent) real tensor. For this family $\braket{\psi_1}{\psi_2}=1/4$ exactly. Table~\ref{tab:singlet-example} reports $\|A_j\|_*$ at $\alpha=\pi/5$ for both cuts, obtained via Corollary~\ref{cor:projector-norm-formula} with the exact Casimir projectors of Remark~\ref{rem:casimir-projectors}; unlike Example~\ref{ex:aligned-singlets}, the values genuinely differ across $j$, and at the cut $S\mid S^c$ the $A_1$ block is a bona fide $3\times3$ matrix (multiplicity $m_1^{(3)}=3$ on both sides) with singular values $(0.6158,0.2171,0.2171)$, not a multiple of the identity.
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\begin{table}[h]
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\centering
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\small
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\begin{tabular}{@{}lcccc@{}}
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\toprule
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cut & $j=0$ & $j=1$ & $j=2$ & $j=3$ \\
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\midrule
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$ABC\mid DEF$ & $0.423725$ & $1.050040$ & $1.634512$ & $1.576275$ \\
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$AB\mid CDEF$ & $0.833607$ & $1.104610$ & $2.064763$ & --- \\
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\bottomrule
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\end{tabular}
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\caption{$\|A_j\|_*$ at $\alpha=\pi/5$ (Example~\ref{ex:singlet-superposition}), both cuts, from exact Casimir projectors; $\sum_j(2j+1)\|A_j\|_*$ matches $\|M\|_*$ computed directly by brute-force simulation to $10^{-6}$ in both cases (reproduced by \texttt{scripts/apply\_exact\_projectors.py}). No $j=3$ block exists at the $2\mid4$ cut since the two-qubit source has $m_j^{(2)}=0$ for $j>2$.}
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\label{tab:singlet-example}
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\end{table}
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\end{example}
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\subsubsection{Cut-independent data: one triple $(T_1,T_2,C_{12})$ for every cut}
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\label{sec:cut-independent-data}
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The point of Eq.~\eqref{eq:coherence-decomposition} is that $T_1,T_2,C_{12}$ do not depend on $\alpha$ \emph{or} on the choice of cut: they are fixed tensors in $(\mathbb R^3)^{\otimes6}$, and every cut's shadow-map block for every $\alpha$ is obtained from them by the fixed linear combination~\eqref{eq:coherence-decomposition} followed by an ordinary reshape, with no further contraction over the full Hilbert space. This was verified directly: brute-force simulation of $\rho_\Xi(\alpha)$ for several $\alpha$, both cuts, matches the prediction from $(T_1,T_2,C_{12})$ to machine precision throughout (\texttt{scripts/recoupling\_check.py}). We record the general statement.
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\begin{proposition}[Cut-independent template data]
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\label{prop:coherence-templates}
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Let $\{\ket{\phi_a}\}_{a=1}^r$ be finitely many pure states, each an exact eigenvector (not merely a stabilizer of the density matrix) of the collective action~\eqref{eq:collective-hypothesis} for a common $G$, i.e.\ $U_g^{\otimes n}\ket{\phi_a}=\ket{\phi_a}$ for every $g\in G$ and every $a$. Define the $r^2$ fixed tensors $T_{ab}(\vec\imath):=\bra{\phi_a}\sigma_{\vec\imath}\ket{\phi_b}$ (so $T_{aa}=T_a$ of Eq.~\eqref{eq:coherence-decomposition} and $T_{ab}+T_{ba}=C_{ab}$ for $a\ne b$). Then for every $\rho=\sum_{a,b}c_{ab}\ketbra{\phi_a}{\phi_b}$ in the span of $\{\ketbra{\phi_a}{\phi_b}\}$ with $\tr\rho=1$, $\rho\ge0$, the full correlation tensor is
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\begin{equation}
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T(\rho) = \sum_{a,b} c_{ab}\,T_{ab},
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\label{eq:general-coherence-decomposition}
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\end{equation}
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independent of which cut's matricization is subsequently taken.
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\end{proposition}
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\begin{proof}
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$T(\rho)(\vec\imath)=\tr(\rho\,\sigma_{\vec\imath})$ is linear in $\rho$; substituting $\rho=\sum_{a,b}c_{ab}\ketbra{\phi_a}{\phi_b}$ and using $\tr(\ketbra{\phi_a}{\phi_b}\sigma_{\vec\imath})=\bra{\phi_b}\sigma_{\vec\imath}\ket{\phi_a}=T_{ba}(\vec\imath)$ gives Eq.~\eqref{eq:general-coherence-decomposition} termwise; the cut dependence enters only afterward, through the reshape of the index set $\vec\imath$, and does not affect the coefficients $c_{ab}$ or the tensors $T_{ab}$.
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\end{proof}
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% TODO (open / not yet carried out): Proposition~\ref{prop:coherence-templates}
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% is stated and used above only for r=2 real generators (psi_1, psi_2), where
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% the single real cross term C_{12} suffices because the c_{ab} happen to be
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% real and symmetric in that example. For general r this requires r(r+1)/2
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% independent real tensors (r diagonal T_aa plus the real/imaginary parts of
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% the r(r-1)/2 off-diagonal overlaps, or equivalently just track the full
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% Hermitian r x r coefficient matrix c_{ab} against the T_{ab} disregarding
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% the a<->b symmetry) -- this is mechanical but has not been written out or
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% numerically tested here for r>2. Do this before submission if the general
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% statement is kept; otherwise restate the Proposition for r=2 only and move
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% the general case to a remark.
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\begin{remark}[What this does and does not establish]
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\label{rem:recoupling-scope}
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Proposition~\ref{prop:coherence-templates} is a bookkeeping consequence of the linearity already used throughout this note (cf.\ the unfolding proposition and the "one rank-one fact, inherited everywhere" theorem of \cite{aschauer2026a}); its content is operational rather than a new inequality: once the $r^2$ template tensors are computed -- from simulation, or in principle extracted from permutationally/collectively-adapted state tomography -- every cut, every mixing angle, and every isotypic block $\|A_j\|_*$ of Corollary~\ref{cor:projector-norm-formula} for the resulting state family is available without revisiting the full $\bigl(\mathbb R^3\bigr)^{\otimes n}$-sized contraction again.
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% TODO (deferred to future work, see session notes): the stronger,
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% state-independent question of whether the reduced blocks A_j themselves
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% (not just the raw tensor) transform between two different cuts of the
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% SAME invariant tensor via an explicit, computable recoupling map -- i.e.
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% a Racah/6j-symbol type formula relating A_j at cut S|S^c directly to the
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% A_{j'} at a different cut S'|S'^c, without reshaping the full raw tensor
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% -- is NOT established here. An exact branching-dimension check (m_p^{(4)}
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% = sum over compatible y of m_y^{(3)} for every p=0,...,4) is consistent
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% with such a relation existing, but the explicit coefficients (presumably
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% governed by an SU(2) recoupling/6j-symbol calculation for the six spin-1
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% legs, or by the Brauer-algebra structure of Inv((R^3)^{\otimes 6}) since
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% the example states here are literally perfect-matching invariants) have
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% not been derived or numerically verified. Left for later.
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\end{remark}
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% TODO (outlook, not attempted): combining the S_m-permutation collapse of
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% Section~\ref{sec:polynomial-witness} (Proposition~\ref{prop:multinomial-collapse})
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% with the SO(3) branching of this subsection. For a cluster invariant under
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% BOTH S_m and collective SO(3) simultaneously, Sym^m(R^3) is known classically
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% to decompose multiplicity-free into spherical harmonics, m_j=1 for
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% j=m,m-2,...; combined with Proposition~\ref{prop:multinomial-collapse} this
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% should collapse a full sector to O(m) one-dimensional channels rather than
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% the polynomial-but-not-linear \binom{m+2}{2} of that Proposition alone.
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% Not worked out or tested here.
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\subsection{Exact degeneracy from stabilizer structure}
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The degeneracies recorded for $\GHZ_3$, the Smolin state, and all $38$ four-qubit graph states share no continuous symmetry of the kind used above. Their common origin is instead a discrete, combinatorial fact about \emph{Pauli-diagonal} states, requiring only elementary group theory over $\mathbb F_2$, and it is this mechanism --- not Proposition~\ref{prop:block-diagonal} --- that is responsible for every degeneracy reported in \cite{aschauer2026a}. That reduced density matrices of stabilizer states are maximally mixed on their support, with the entanglement across any cut given by the rank of the corresponding restriction of the stabilizer group, is a classical fact \cite{fattal2004stabilizer}; Lemma~\ref{lem:code-support} below is the flat-support statement underlying that result, and Lemma~\ref{lem:stabilizer-degeneracy} translates it directly into the singular-value structure of the shadow map itself, rather than into an entanglement entropy.
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