feat: add new scripts for S_m x SO(3) double collapse examples and common utilities

paper/shadow_maps_symmetric_states.tex

@@ -527,7 +527,7 @@ numerically identical to the cut bound in \cite{aschauer2026a}, but now a statem
 \subsection{Axial $U(1)$ symmetry: an exact worked example}
 \label{sec:u1-example}
 
-The isotypic mechanism of Proposition~\ref{prop:block-diagonal} is easiest to see concretely for an abelian symmetry group, where it reduces to an ordinary charge-conservation selection rule. That a global $U(1)$ (or other abelian) symmetry splits the singular values of a bipartite correlation-type matrix into charge sectors is by now a recurring theme in the entanglement-detection literature under the name \emph{symmetry-resolved entanglement}: it has been used to sharpen partial-transpose-moment witnesses \cite{nevenetal2021}, to decompose the singular values of the realignment matrix itself into $U(1)$-charge blocks \cite{rathetal2023}, and, in a very recent and structurally close construction, to build symmetry-resolved bounds on the negativity directly from charge-sector-resolved realignment-matrix singular values \cite{tarabungahaug2025}. The mechanism below is the same charge-conservation selection rule applied instead to the bigraduated shadow map $\mathcal M_S(\rho)$, worked out completely and in closed form for one illustrative four-qubit family rather than as a general witness-improvement scheme.
+The isotypic mechanism of Proposition~\ref{prop:block-diagonal} is easiest to see concretely for an abelian symmetry group, where it reduces to an ordinary charge-conservation selection rule. That a global $U(1)$ (or other abelian) symmetry splits the singular values of a bipartite correlation-type matrix into charge sectors is by now a recurring theme in the entanglement-detection literature under the name \emph{symmetry-resolved entanglement}: it has been used to sharpen partial-transpose-moment witnesses \cite{nevenetal2021}, to decompose the singular values of the realignment matrix itself into $U(1)$-charge blocks \cite{rathetal2023}, and, in a very recent and structurally close construction, to build symmetry-resolved bounds on the negativity directly from charge-sector-resolved realignment-matrix singular values \cite{tarabungahaug2025}. That literature block-diagonalizes the reduced state $\rho_A$ itself (or a moment thereof) and works almost exclusively with abelian charges; Proposition~\ref{prop:block-diagonal} instead block-diagonalizes the linear \emph{response map} $\mathcal M_S(\rho)$, and, in Section~\ref{sec:full-su2-symmetry} below, does so for the non-abelian group $SU(2)$ via the Casimir operator rather than a conserved charge. The mechanism below is the same charge-conservation selection rule applied instead to the bigraduated shadow map $\mathcal M_S(\rho)$, worked out completely and in closed form for one illustrative four-qubit family rather than as a general witness-improvement scheme.
 
 \begin{remark}[Abelian symmetries as selection rules]
 \label{rem:abelian-selection-rule}
@@ -603,6 +603,9 @@ This example illustrates Corollary~\ref{cor:joint-refinement} concretely: the fu
 % ============================================================
 
 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,
+
+The representation-theoretic content of what follows --- Casimir diagonalization in place of group averaging, and Wigner--Eckart reduction of a tensor to a Clebsch--Gordan coefficient times a smaller reduced operator --- is not new machinery on its own. It is standard practice under the name \emph{spin-adapted DMRG} in ab-initio quantum chemistry, where exploiting the non-abelian $SU(2)$ spin symmetry of the electronic Hamiltonian via exactly this Wigner--Eckart factorization \cite{sharmachan2012spinadapted} routinely enables density-matrix-renormalization-group calculations on active spaces with well over a hundred orbitals, e.g.\ CAS$(113,76)$ on GPU hardware \cite{menczer2024quarterpetaflops}. We are not aware of this toolbox having been carried through as systematically for the entanglement-witness setting of this note, where the reduced object of interest is a witness-relevant response map rather than a Hamiltonian or an MPS tensor; the two settings share the group theory but differ in what is being reduced and why. A related but distinct use of collective-unitary symmetry appears in \cite{huberklepmagronvolcic2022werner}, which constructs dimension-free entanglement witnesses for Werner states (invariant under the diagonal conjugate action of the full unitary group, not merely collective single-qubit rotation) via a semidefinite-programming hierarchy; the guiding idea --- exploit a fixed symmetry to obtain a structural rather than dimension-dependent result --- is the same in spirit, but the symmetry group, the object being reduced, and the resulting statement (existence of a witness, versus the block structure of a given response map) are all different.
+
 \begin{equation}
 g\ \longmapsto\ \bigotimes_{a\in P}U_g,
 \qquad
@@ -860,7 +863,7 @@ satisfying $3A_1+7A_3=1+14=15=\norm{M_{S\to S^c}(\ketbra QQ)}_*$ exactly, matchi
 
 \subsection{Exact degeneracy from stabilizer structure}
 
-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.
+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}; the same $\mathbb F_2$-linear stabilizer-restriction toolbox underlies the entanglement witnesses of Tóth and Gühne \cite{tothguhne2005stabilizer}, though their target object is a witness operator rather than the singular-value structure of a response map. 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 or a witness.
 
 \paragraph{Setup.} Identify each single-party Pauli index with $\mathbb F_2^2$ via $I\mapsto(0,0)$, $X\mapsto(1,0)$, $Y\mapsto(1,1)$, $Z\mapsto(0,1)$, so that an $n$-party Pauli string $\sigma_{\vec i}$ corresponds to $\vec i\in\mathbb F_2^{2n}$, and string multiplication (up to phase) becomes addition. Let $H\le\mathbb F_2^{2n}$ be an isotropic subgroup (i.e.\ its elements pairwise commute as operators) not containing $-I$, and let
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