Merge section on symmetric states

paper/symmetric_shadow_maps_formal.tex

@@ -16,6 +16,7 @@
 \newtheorem{definition}{Definition}
 \newtheorem{corollary}{Corollary}
 \newtheorem{remark}{Remark}
+\newtheorem{lemma}{Lemma}
 
 \newcommand{\tr}{\operatorname{tr}}
 \newcommand{\id}{\mathbbm{1}}
@@ -67,8 +68,6 @@ They are local-unitary invariants and admit a clean convexity-based entanglement
 
 The point of the present note is that the old geometric picture has a natural multipartite continuation. Instead of compressing each block to one scalar, we keep the full family of source-indexed correlation-response maps produced by a chosen source party and stack the responses landing in the different orthogonal sectors of the complement. In this sense the main contribution is best viewed as a new object architecture rather than a new functional applied to a familiar full correlation tensor. The general construction and cut-separable bound below work for arbitrary finite local dimensions; the later biseparable benchmark and all numerical examples then specialize to qubits, where the constants and geometry are especially transparent.
 
-%% A substantial literature already studies separability criteria in the Bloch-representation and correlation-tensor language, including bipartite correlation-matrix criteria \cite{devicente,chenwu}, multipartite full-tensor unfoldings and matricizations \cite{hassanjoag,devicentehuber,li2014,jingzhang2023}, nonlinear geometric tensor criteria \cite{laskowski2011}, scalar multi-sector norm criteria \cite{klocklhuber2015}, and more recent augmented or partition-adapted mixed-order block constructions \cite{shen2016,sarbicki2020,zhao2020,huang2024extended,liyaoyangfei2025}. This note is intended to sit explicitly within that landscape rather than outside it.
-
 A substantial literature already studies separability criteria in the Bloch-representation and correlation-tensor language, including bipartite correlation-matrix criteria \cite{devicente,chenwu}, multipartite unfoldings and matricizations of the full correlation tensor \cite{hassanjoag,devicentehuber,li2014,jingzhang2023}, nonlinear geometric tensor criteria \cite{laskowski2011}, scalar multi-sector norm criteria \cite{klocklhuber2015}, and more recent extended or partition-adapted mixed-order block constructions \cite{shen2016,sarbicki2020,zhao2020,huang2024extended,liyaoyangfei2025}. This note is intended to sit explicitly within that landscape rather than outside it.
 
 Historically, it is useful to distinguish two nearby lines of work. The local-invariant sector decomposition of Aschauer \emph{et al.} \cite{aschauer} already gave an early multipartite entanglement criterion in terms of the coefficients of a local operator expansion; that expansion is carried out, from the first equations of that paper onward, in the full product basis of local generators together with the identity on each party, so the full tensor $\mathcal C(\rho)$ used in Section~\ref{sec:tensor-viewpoint} below is already present there, and the sector-restricted tensor $C_S(\rho)$ used for $L_S$ is obtained from it by exactly the restriction to nonidentity indices already carried out in that paper. Later work such as Hassan and Joag \cite{hassanjoag} made the Bloch-representation terminology explicit and developed criteria from full-tensor unfoldings. The present construction belongs to that broader correlation-tensor lineage, but its organization is closest in spirit to, and its starting tensor is literally the one already used in, \cite{aschauer}.
@@ -146,7 +145,7 @@ be the orthogonal projection. The sector response maps are then simply the proje
 \begin{equation*}
 \widetilde M_{a\to T}(\rho):=P_T\widetilde{\mathcal M}_a(\rho).
 \end{equation*}
-After choosing orthonormal bases in $\V_0^{(a)}$ and in each $\V_T^{(\bar a)}$, these become the coordinate maps used below. We will use the shorter term \emph{shadow maps} for this family.
+After choosing orthonormal bases in $\V_0^{(a)}$ and in each $\V_T^{(\bar a)}$, these become the coordinate maps used below. We will use the shorter term \emph{shadow maps} for this family. We emphasize that this usage is unrelated to the classical-shadows measurement protocols of \cite{huangkuengpreskill2020}: the shadow maps of this note are linear response operators built directly from the correlation tensor of $\rho$, not estimators reconstructed from randomized single-copy measurements.
 
 \begin{definition}
 \label{def:combined-shadow}
@@ -657,16 +656,6 @@ For each party $a$, we augment the local index range by including $i_a = 0$, whi
 \end{equation*}
 \end{definition}
 
-%% \begin{definition}[Augmented Bloch tensor]
-%% \label{def:augmented-tensor}
-%% For each party $a$, extend the index range of the local generators to $i_a\in\{0,1,\dots,d_a^2-1\}$, so that $i_a=0$ selects the identity $\sigma_0^{(a)}=\id$ already introduced in Section~\ref{sec:old-criterion}. Define the order-$n$ tensor
-%% \begin{equation*}
-%% \mathcal C(\rho)\in\bigotimes_{a\in P}\R^{d_a^2},
-%% \qquad
-%% \bigl(\mathcal C(\rho)\bigr)_{i_1,\dots,i_n}=\tr\!\Bigl(\rho\bigotimes_{a\in P}\sigma^{(a)}_{i_a}\Bigr).
-%% \end{equation*}
-%% \end{definition}
-
 Since $\bigotimes_{a\in P}\R^{d_a^2}=\bigotimes_{a\in P}\bigl(\R\oplus\R^{d_a^2-1}\bigr)$ expands by distributivity into $2^n$ orthogonal summands indexed by which legs are trivial, every sector tensor of Eq.~\eqref{eq:corr-tensor-def} is simply a slice of $\mathcal C(\rho)$:
 \begin{equation}
 C_V(\rho)=\mathcal C(\rho)\big|_{\,i_a\neq0\text{ for }a\in V,\ i_a=0\text{ for }a\notin V},
@@ -784,13 +773,8 @@ Thus the state is PPT up to numerical precision, so the Peres-Horodecki PPT test
 
 \subsection*{Qubit Pauli-tensor form}
 
-%% For qubits we use the convention already anticipated in the Smolin example of Section~\ref{sec:multiparty-sources}: $\sigma_0=\id$ and $\sigma_1=X$, $\sigma_2=Y$, $\sigma_3=Z$. The augmented tensor is then the full Pauli-correlation tensor
-
 For qubits we use the convention already anticipated in the Smolin example of Section~\ref{sec:multiparty-sources}: $\sigma_0=\id$ and $\sigma_1=X$, $\sigma_2=Y$, $\sigma_3=Z$. In this notation, the full Bloch tensor $\mathcal C(\rho)$ becomes the familiar Pauli-correlation tensor
 
-%\subsection*{Qubit Pauli-tensor form}
-
-%For qubits, take $\sigma_0=\id$ and $\sigma_1=X$, $\sigma_2=Y$, $\sigma_3=Z$. The augmented tensor is then the full Pauli-correlation tensor
 \begin{equation*}
 \mathcal C(\rho)_{i_1,\dots,i_n}
 =
@@ -852,6 +836,33 @@ A fully separable state is separable across every one-vs-rest cut. The claim fol
 
 In equal local dimensions these functionals are permutation invariant. More generally, they are source-aggregated cut-sensitive scalars. The average probes all one-vs-rest cuts simultaneously, whereas the maximum asks whether at least one cut exhibits a large combined shadow ellipsoid.
 
+This averaging-over-cuts strategy for turning a bipartite-type bound into a
+genuine multipartite criterion is not unique to the present construction. A
+structurally analogous move appears in \cite{liyaoyangfei2025}, whose
+Theorem~5 averages the trace norm of a bipartition-indexed extended
+correlation tensor over \emph{all} $2^{N-1}-1$ bipartitions of the $N$
+parties, comparing the result against a bound assembled from the worst case
+over bipartitions of each size $k=1,\ldots,N-1$, to certify genuine
+$N$-partite entanglement. Three differences are worth making explicit.
+First, the sum in \cite{liyaoyangfei2025} runs over all $2^{N-1}-1$
+bipartitions of $P$, including every $k$-vs-$(N-k)$ split with
+$1<k<N-1$, whereas $\Phi_{\mathrm{sym}}$ and $\Phi_{\max}$ here range only
+over the $n$ one-vs-rest cuts; the two index sets coincide at $n=3$ but
+diverge from $n=4$ on, where the bipartition sum there also includes, e.g.,
+the $2\mid2$ splits that a one-vs-rest average does not see directly.
+Second, only the averaged quantity is ever compared against a bound in
+\cite{liyaoyangfei2025}; the maximum that appears there acts solely on the
+threshold side (the worst case per partition size, entering the bound
+$M_0^{(1\cdots N)}$), not as a second, independently observed witness the
+way $\Phi_{\max}(\rho)$ is here. Third, and more relevant to the underlying
+object, each per-bipartition term in \cite{liyaoyangfei2025} collapses both
+sides of the cut to their single joint correlation tensor before the trace
+norm is taken, so no sector structure survives inside either half of the
+partition. $\mathcal M_a(\rho)$, by contrast, is a direct sum over every
+nonempty subset $T$ of the complement $\bar a$; it is this retained sector
+granularity on one side of the cut, rather than the averaging step itself,
+that we take to be the substantive point of the present construction.
+
 \subsection*{Three-qubit biseparable benchmark}
 
 For three qubits the symmetric average can be optimized explicitly over the biseparable set. This is the first place where we use genuinely qubit-specific formulas rather than only the general response-map architecture. Suppose first that
@@ -957,32 +968,6 @@ p>\frac{\sqrt 7}{3}\approx 0.882,
 so this example is much more fragile under white noise than the graph-state families discussed below.
 
 
-
-%% \subsection*{The four-qubit Smolin state}
-%% 
-%% For the Smolin state,
-%% \begin{equation*}
-%% \rho_{\mathrm{Smo}}=\frac1{16}\left(\id^{\otimes 4}+X^{\otimes 4}+Y^{\otimes 4}+Z^{\otimes 4}\right),
-%% \end{equation*}
-%% all one-, two-, and three-body correlations vanish. Only the full four-body sector survives. Thus for each source party $a$ the combined map has three orthogonal response directions and singular values
-%% \begin{equation*}
-%% \frac{1}{\sqrt 7},\qquad \frac{1}{\sqrt 7},\qquad \frac{1}{\sqrt 7},
-%% \end{equation*}
-%% because the normalization factor is now $\sqrt{2^3-1}=\sqrt 7$. Hence
-%% \begin{equation*}
-%% \norm{\mathcal M_a(\rho_{\mathrm{Smo}})}_* = \frac{3}{\sqrt 7}>1,
-%% \end{equation*}
-%% and therefore
-%% \begin{equation*}
-%% \Phi_{\mathrm{sym}}(\rho_{\mathrm{Smo}})=\Phi_{\max}(\rho_{\mathrm{Smo}})=\frac{3}{\sqrt 7}>1.
-%% \end{equation*}
-%% So the criterion correctly detects entanglement across every $1\mid 3$ cut. At the same time, this is not a genuine-multipartite conclusion, since the Smolin state is separable across every $2\mid 2$ split.
-%% For the white-noise family $p\rho_{\mathrm{Smo}}+(1-p)\id/16$, the fully separable threshold would only be crossed for
-%% \begin{equation*}
-%% p>\frac{\sqrt 7}{3}\approx 0.882,
-%% \end{equation*}
-%% so this example is much more fragile under white noise than the graph-state families discussed below.
-
 \subsection*{A numerical four-qubit graph-state scan}
 
 Using the \texttt{qtensor} package, we also evaluated the symmetric shadow functionals for all connected labeled graph states on four qubits. Numerically, all $38$ such graph states give the same value,
@@ -997,7 +982,7 @@ the fully separable threshold is crossed already at
 \begin{equation*}
 p>\frac{\sqrt 7}{6}\approx 0.441.
 \end{equation*}
-This includes the canonical examples $\GHZ_4$, the line graph state, the ring graph state, and the star graph state.
+This includes the canonical examples $\GHZ_4$, the line graph state, the ring graph state, and the star graph state \cite{heinEisertBriegel2004}.
 
 The same numerical scan shows that this shadow detection is not explained by pairwise entanglement in the reduced states. For those representative families, every two-qubit marginal remains PPT at the threshold $p=\sqrt 7/6$, and for the line and ring graph states some of the two-qubit marginals are even maximally mixed. So the shadow functional is responding to multipartite correlation structure that is not visible in pair reductions. This is still not a four-qubit genuine-multipartite-entanglement proof, because the corresponding biseparable threshold is not yet known, but it makes the criterion promising as a genuinely multipartite diagnostic.
 
@@ -1026,6 +1011,256 @@ To probe the unresolved four-qubit biseparable benchmark, we performed a small r
 \Phi_{\mathrm{sym}}\approx 2.235
 \end{equation*}
 for a state separable across a $2\mid 2$ partition, while the best sampled $1\mid 3$ values were only around $1.94$. This is not a proof of the true biseparable threshold, but it suggests two useful heuristics: first, the most dangerous competitors to the graph-state value $6/\sqrt 7\approx 2.268$ come from $2\mid 2$ cuts rather than $1\mid 3$ cuts; second, the connected four-qubit graph-state value sits slightly above the best random biseparable samples we found.
+\section{Symmetry-adapted block decomposition of the shadow map}
+\label{sec:symmetry-blocks}
+
+The constructions of Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sources} treat the sector grading $V\subseteq S$, $T\subseteq S^c$ as the only available organizing structure on either side of a cut. When $\rho$ carries an additional symmetry --- exactly, or after an appropriate twirl --- this grading can be refined further, along representation-theoretic rather than combinatorial lines. The refinement is worth having for four distinct reasons, which we state before developing the formal statements, since they motivate different parts of what follows and are not equally strong.
+
+\begin{itemize}
+\item \emph{Explanatory power.} Several numerical facts already reported in this note --- the triple degeneracy of the singular values of $\mathcal M_a$ for $\GHZ_3$ (Section~\ref{sec:qubit-numerics}), the triple degeneracy at the Smolin state under both the $1\mid3$ and $2\mid2$ cuts (Section~\ref{sec:multiparty-sources}), and the rigid common value across all $38$ four-qubit graph states --- have so far been recorded as numerical observations. The block-diagonality and stabilizer-support results below show that such degeneracies are not coincidental: they are forced exactly, by two complementary mechanisms depending on whether $\rho$ is symmetry-invariant in a representation-theoretic sense or Pauli-diagonal in a stabilizer sense. This converts a family of separately verified numerical facts into structural statements.
+\item \emph{Diagnostic resolution.} The sub-block witnesses of Corollary~\ref{cor:sub-block} resolve a violation only down to the level of \emph{which parties} are involved ($V,T$). A representation-theoretic decomposition, where applicable, resolves it further, down to \emph{which symmetry channel} within a fixed $(V,T)$ sector is responsible --- for instance, whether a two-party source correlation block is carrying its signal in a totally symmetric or in an antisymmetric combination of its constituents. This is invisible to the party-indexed grading alone.
+\item \emph{Computational cost.} For a source or target cluster respecting a symmetry group $G$, an isotypic decomposition replaces one singular value decomposition on the full sector space by several independent, much smaller singular value decompositions on the multiplicity spaces $M_\lambda$, whose dimensions grow far more slowly than the ambient sector dimension as the cluster size increases. This is the natural computationally tractable foothold for the symmetric sub-family of the higher-order tensor construction discussed in the closing remark of Section~\ref{sec:tensor-viewpoint}.
+\item \emph{A one-sided extension via twirling.} If $\rho$ itself lacks the relevant symmetry but a twirl $T_G(\rho)$ is cheap to evaluate, entanglement detected on $T_G(\rho)$ certifies entanglement of $\rho$ (twirling by local unitaries preserves separability), so a block-diagonal criterion can serve as an inexpensive pre-test, with the standard one-sided caveat that a negative result on $T_G(\rho)$ is uninformative about $\rho$.
+\end{itemize}
+
+What this refinement does \emph{not} deliver is a numerically sharper detection threshold: Remark~\ref{rem:no-universal-sharpening} below shows that no universal, state-independent improvement over Theorems~\ref{thm:cut-bound} and \ref{thm:cluster-cut} exists at the level of an individual representation-theoretic block, except in the single-isotype case where the improvement is one of concentration rather than of threshold value.
+
+\subsection*{Setup}
+
+Let $G$ be a compact group acting on the system by local unitaries, $g\mapsto \bigotimes_{a\in P}U_g^{(a)}$, and suppose $\rho$ is $G$-invariant: $(\bigotimes_a U_g^{(a)})\,\rho\,(\bigotimes_a U_g^{(a)})^\dagger=\rho$ for all $g\in G$. Fix a cut $S\mid S^c$ preserved by $G$ as a set partition, so that $G$ acts on $\V_0^{(S)}$ and on $\V_0^{(S^c)}$ separately, via the adjoint representations $\mathrm{Ad}^{(S)}_g$, $\mathrm{Ad}^{(S^c)}_g$. Decompose both traceless spaces into isotypic components,
+\begin{equation}
+\V_0^{(S)}=\bigoplus_\lambda M_\lambda^{(S)}\otimes V_\lambda,
+\qquad
+\V_0^{(S^c)}=\bigoplus_\lambda M_\lambda^{(S^c)}\otimes V_\lambda,
+\label{eq:isotypic-decomp}
+\end{equation}
+where $\lambda$ ranges over the irreducible representations of $G$ appearing on either side, $V_\lambda$ denotes a fixed model of the irreducible representation of dimension $d_\lambda$, and $M_\lambda^{(S)}$, $M_\lambda^{(S^c)}$ are the corresponding multiplicity spaces. This is exactly the same type of decomposition already used for the source and target sector gradings in Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sources}, now taken with respect to a representation-theoretic rather than a combinatorial grading; the two coincide only in special cases.
+
+The general strategy --- twirl a state over a symmetry group and use Schur's lemma to collapse the resulting computation onto the much smaller multiplicity spaces --- is the same one used classically to simplify the computation of entanglement measures for symmetric states \cite{vollbrechtwerner2001}; the present section applies it to the shadow map itself rather than to a scalar entanglement measure.
+
+\begin{proposition}[Exact block-diagonality]
+\label{prop:block-diagonal}
+Under the hypotheses above, $\widetilde{\mathcal M}_S(\rho)$ is block diagonal with respect to the isotypic decomposition~\eqref{eq:isotypic-decomp}: writing $\Pi_\lambda^{(S^c)}$, $\Pi_\mu^{(S)}$ for the isotypic projections,
+\begin{equation*}
+\Pi_\lambda^{(S^c)}\,\widetilde{\mathcal M}_S(\rho)\,\Pi_\mu^{(S)}=0
+\qquad\text{whenever }\lambda\neq\mu,
+\end{equation*}
+exactly, not merely approximately or in a bound. Moreover, on the surviving diagonal blocks,
+\begin{equation}
+\widetilde{\mathcal M}_S(\rho)\big|_\lambda = A_\lambda\otimes\mathrm{id}_{V_\lambda}
+\label{eq:wigner-eckart}
+\end{equation}
+for a unique linear map $A_\lambda:M_\lambda^{(S)}\to M_\lambda^{(S^c)}$, the reduced shadow map at $\lambda$.
+\end{proposition}
+
+\begin{proof}
+$G$-invariance of $\rho$ gives $\mathrm{Ad}^{(S^c)}_g\,\widetilde{\mathcal M}_S(\rho)=\widetilde{\mathcal M}_S(\rho)\,\mathrm{Ad}^{(S)}_g$ for all $g$, i.e.\ $\widetilde{\mathcal M}_S(\rho)$ is a $G$-equivariant map between the two representations~\eqref{eq:isotypic-decomp}. Both claims are then Schur's lemma applied to the isotypic decomposition: equivariant maps vanish between inequivalent irreducible summands, and act as a fixed scalar multiple of the identity on the irreducible factor $V_\lambda$ within a matching pair, leaving exactly the freedom recorded in $A_\lambda$ on the multiplicity spaces.
+\end{proof}
+
+Proposition~\ref{prop:block-diagonal} organizes $\widetilde{\mathcal M}_S(\rho)$ by symmetry channel alone, treating $\V_0^{(S)}$ and $\V_0^{(S^c)}$ as undifferentiated representations of $G$. But these spaces already carry the combinatorial sector grading of Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sources}, $\V_0^{(S)}=\bigoplus_{\emptyset\neq V\subseteq S}\V_V^{(S)}$. The next result shows that this grading and the isotypic one are never in tension: for any symmetry acting locally by conjugation, one refines the other, and both can be imposed simultaneously without contradiction.
+
+\begin{proposition}[Compatibility of sector and symmetry decompositions]
+\label{prop:sector-symmetry-compatible}
+Let $G$ act on the system by local unitaries, $g\mapsto\bigotimes_{a\in P}U_g^{(a)}$, and let $S\subseteq P$. Writing $R(g):=\bigotimes_{a\in S}\mathrm{Ad}_{U_g^{(a)}}$ for the induced action on $\V^{(S)}$, we have
+\begin{equation}
+R(g)\,\V_V^{(S)} = \V_V^{(S)}
+\qquad\text{for every } g\in G \text{ and every nonempty } V\subseteq S.
+\label{eq:sector-invariance}
+\end{equation}
+Consequently the sector projectors $P_V$ of Eq.~\eqref{eq:source-bloch-decomposition} commute with $R(g)$ for every $g\in G$, hence with every isotypic projector $P_\lambda=d_\lambda\int_G\chi_\lambda(g)^*R(g)\,dg$ of Eq.~\eqref{eq:isotypic-decomp}, and $\V^{(S)}$ decomposes simultaneously as
+\begin{equation}
+\V^{(S)} = \bigoplus_{\emptyset\neq V\subseteq S}\ \bigoplus_\lambda \V_{V,\lambda},
+\qquad
+\V_{V,\lambda}:=P_VP_\lambda\V^{(S)}.
+\label{eq:joint-decomposition}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Since $\mathrm{Ad}_{U_g^{(a)}}$ fixes the identity, $\mathrm{Ad}_{U_g^{(a)}}\id^{(a)}=\id^{(a)}$, and maps the traceless Hermitian subspace $\V_0^{(a)}$ to itself (conjugation by a unitary preserves both Hermiticity and tracelessness), each local factor space splits $G$-invariantly as $\mathbb C\,\id^{(a)}\oplus\V_0^{(a)}$. The sector space $\V_V^{(S)}$ is, by definition, exactly the span of product basis elements with the $a$-th factor in $\V_0^{(a)}$ for $a\in V$ and equal to $\id^{(a)}$ for $a\notin V$. Since $R(g)$ acts factor-wise and each local factor preserves its own identity-versus-traceless split, $R(g)$ cannot move an element of $\V_V^{(S)}$ out of $\V_V^{(S)}$: it maps trivial legs to trivial legs and active legs to (possibly rotated, but still traceless) active legs, without ever changing which legs are trivial. This gives $R(g)\V_V^{(S)}\subseteq\V_V^{(S)}$; the same argument applied to $g^{-1}$ gives the reverse inclusion, hence Eq.~\eqref{eq:sector-invariance}. Commutation of $P_V$ with $R(g)$, and therefore with the integral defining $P_\lambda$, follows immediately.
+\end{proof}
+
+\begin{corollary}[Joint refinement of the bigraduated shadow map]
+\label{cor:joint-refinement}
+Let $S\mid S^c$ be a cut preserved by $G$ as a set partition, with $G$ acting locally by conjugation on both $S$ and $S^c$ as above. Then, whenever $\rho$ is $G$-invariant, every bigraduated block $M_{V\to T}(\rho)$ of Definition~\ref{def:bigraduated} further decomposes as
+\begin{equation}
+M_{V\to T}(\rho) = \bigoplus_\lambda I_{d_\lambda}\otimes A_{V,T,\lambda}(\rho),
+\qquad \emptyset\neq V\subseteq S,\ \ \emptyset\neq T\subseteq S^c,
+\label{eq:joint-block}
+\end{equation}
+for reduced maps $A_{V,T,\lambda}$ on the corresponding multiplicity spaces. In particular, the combinatorial sector grading of Section~\ref{sec:multiparty-sources} and the representation-theoretic isotypic grading of the present section are not competing organizations of $\mathcal M_S(\rho)$, but two orthogonal refinements of the same operator, and can be applied jointly: one may first restrict to a sub-block witness $M_{V\to T}$ as in Corollary~\ref{cor:sub-block} and then further resolve it by symmetry channel, or apply the two refinements in the opposite order, with the same result.
+\end{corollary}
+
+\begin{proof}
+By Proposition~\ref{prop:sector-symmetry-compatible} applied on the source side to $\V_V^{(S)}$ and, symmetrically, on the target side to $\V_T^{(S^c)}$, the restrictions $R_S(g)\big|_{\V_V^{(S)}}$ and $R_{S^c}(g)\big|_{\V_T^{(S^c)}}$ are themselves well-defined representations of $G$. Since $\rho$ is $G$-invariant, $\widetilde{\mathcal M}_S(\rho)$ intertwines $R_S(g)$ and $R_{S^c}(g)$ on the full spaces (Proposition~\ref{prop:block-diagonal}), and since the sector inclusion $\iota_V$ and sector projection $P_T$ used in Definition~\ref{def:bigraduated} are themselves $G$-equivariant by Proposition~\ref{prop:sector-symmetry-compatible}, the composite $M_{V\to T}(\rho)=P_T\,\widetilde{\mathcal M}_S(\rho)\,\iota_V$ intertwines the restricted representations on $\V_V^{(S)}$ and $\V_T^{(S^c)}$. Schur's lemma applied to this restricted intertwiner gives Eq.~\eqref{eq:joint-block}.
+\end{proof}
+
+\subsection*{Exact formula and its consequence}
+
+\begin{proposition}[Reduced matrix element formula]
+\label{prop:reduced-formula}
+Let $\rho=T_G(\rho_S\otimes\sigma_{S^c})$ be the $G$-twirl of a product state across $S\mid S^c$, so $\rho$ is separable (a mixture, over $g\in G$, of product states) and $G$-invariant. Let $r^{(S)}\in\V_0^{(S)}$, $v_{S^c}\in\V_0^{(S^c)}$ be the traceless Bloch vectors of $\rho_S$, $\sigma_{S^c}$, with isotypic components $r_\lambda$, $v_\lambda$. Then
+\begin{equation}
+A_\lambda=\frac{1}{d_\lambda}\,\tilde v_\lambda\,\tilde r_\lambda^{\,T},
+\label{eq:reduced-element}
+\end{equation}
+where $\tilde r_\lambda$, $\tilde v_\lambda$ are $r_\lambda$, $v_\lambda$ reshaped as $(\dim M_\lambda^{(S)})\times d_\lambda$ and $(\dim M_\lambda^{(S^c)})\times d_\lambda$ matrices in a basis of $V_\lambda$ shared by both sides. Consequently,
+\begin{equation}
+\dim(V_\lambda)\,\norm{A_\lambda}_*\;\le\;\norm{r_\lambda}\,\norm{v_\lambda}.
+\label{eq:per-block-bound}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+For fixed $g$, the product term $(\mathrm{Ad}^{(S)}_g r^{(S)})(\mathrm{Ad}^{(S^c)}_g v_{S^c})^T$ (unnormalized) is rank one, exactly as in the proof of Theorem~\ref{thm:cluster-cut}. Averaging over $g$ and expanding both factors in the isotypic bases gives, by the Schur orthogonality relation $\int_G D^\lambda(g)_{cb}D^\lambda(g)_{da}\,dg=\tfrac1{d_\lambda}\delta_{cd}\delta_{ab}$ for the (real, orthogonal) irreducible matrix elements $D^\lambda$, exactly Eq.~\eqref{eq:reduced-element} on each isotypic block, with all cross-$\lambda$ contributions vanishing by the same orthogonality relation applied to inequivalent irreducibles. Equation~\eqref{eq:per-block-bound} then follows from the standard nuclear-norm bound $\norm{AB}_*\le\norm{A}_\fro\norm{B}_\fro$ applied to Eq.~\eqref{eq:reduced-element}, using $\norm{\tilde r_\lambda}_\fro=\norm{r_\lambda}$, $\norm{\tilde v_\lambda}_\fro=\norm{v_\lambda}$.
+\end{proof}
+
+\begin{corollary}[Consistency with the cut-separable bound]
+\label{cor:consistency}
+Under the hypotheses of Proposition~\ref{prop:reduced-formula},
+\begin{equation}
+\sum_\lambda \dim(V_\lambda)\,\norm{A_\lambda}_*
+\;\le\;
+\norm{r^{(S)}}\,\norm{v_{S^c}}
+\;\le\;
+\sqrt{(d_S-1)(d_{S^c}-1)},
+\label{eq:cs-recovery}
+\end{equation}
+the first inequality by Cauchy--Schwarz over $\lambda$ applied to Eq.~\eqref{eq:per-block-bound}, and the second by the correlation-sum identity used in Theorem~\ref{thm:cluster-cut}. By linearity and convexity of the nuclear norm, Eq.~\eqref{eq:cs-recovery} extends to arbitrary $G$-invariant separable states (finite or continuous mixtures of twirled product terms), recovering the bound of Theorem~\ref{thm:cluster-cut} through the block decomposition rather than around it.
+\end{corollary}
+
+\begin{remark}[No universal per-block sharpening, except by concentration]
+\label{rem:no-universal-sharpening}
+The first inequality in Eq.~\eqref{eq:cs-recovery} is generically strict: equality in Cauchy--Schwarz requires $\norm{r_\lambda}\propto\norm{v_\lambda}$ across all $\lambda$, which independently chosen $\rho_S,\sigma_{S^c}$ have no reason to satisfy. A direct numerical check on a twirled random product state gives $\norm{\mathcal M_S(\rho)}_*=0.429$ against the bound $\norm{r^{(S)}}\norm{v_{S^c}}=0.750$ from Eq.~\eqref{eq:cs-recovery} --- a strict, and generic, gap. Since Eq.~\eqref{eq:per-block-bound} bounds $\norm{r_\lambda}$ only by the global $\norm{r^{(S)}}^2\le d_S-1$, with no constraint on how the purity budget distributes across $\lambda$ for a generic $\rho_S$, no universal constant improving on $\sqrt{(d_S-1)(d_{S^c}-1)}$ holds for an individual block $\lambda$ in general.
+
+The exception is the case where $\V_0^{(S)}$ itself carries only a single isotypic component under $G$: then $r_\lambda=r^{(S)}$ trivially, the full purity budget sits in the one available block, and Eq.~\eqref{eq:per-block-bound} becomes
+\begin{equation*}
+\dim(V_\lambda)\,\norm{A_\lambda}_*\;\le\;\sqrt{(d_S-1)(d_{S^c}-1)},
+\end{equation*}
+numerically identical to Theorem~\ref{thm:cut-bound}, but now a statement about a matrix of size $\dim M_\lambda^{(S^c)}\times1$ rather than the full target space. This single-isotype case is a genuine sharpening of concentration, not of threshold, and applies whenever $\rho$ is actually $G$-invariant for a group under which the source side is irreducible. It does \emph{not}, however, apply to the two states used to motivate this section: $\GHZ_3$ and the Smolin state carry no continuous collective symmetry (a direct check shows both fail to be invariant already under a one-parameter collective rotation), so the representation-theoretic mechanism above is not the explanation for their observed degeneracies. The correct explanation for those two states, and more generally for any Pauli-diagonal state, is combinatorial rather than representation-theoretic, and is developed next.
+\end{remark}
+
+\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.
+
+\begin{remark}[Abelian symmetries as selection rules]
+\label{rem:abelian-selection-rule}
+Let $G=U(1)$ act by collective $z$-rotation, $g=\theta\mapsto\bigotimes_{a\in P}R_z^{(a)}(\theta)$ with $R_z(\theta)=e^{-i\theta Z/2}$. Every irreducible representation of $U(1)$ is one-dimensional, so the isotypic decomposition of Proposition~\ref{prop:block-diagonal} coincides exactly with the eigenspace decomposition of the conserved total charge $S_z^{\mathrm{tot}}$. Writing $E_\pm=(X\pm iY)/\sqrt2$ for the weight-$(\pm1)$ combinations and $E_0=Z$ for the weight-$0$ direction, so that $\mathrm{Ad}_{R_z(\theta)}E_\pm=e^{\pm i\theta}E_\pm$ and $\mathrm{Ad}_{R_z(\theta)}E_0=E_0$, $G$-invariance of $\rho$ forces
+\begin{equation*}
+\tr\bigl(\rho\,\sigma_{i_S}\otimes\sigma_{i_T}\bigr)
+=\tr\bigl(\rho\,\mathrm{Ad}_{R_z(\theta)}(\sigma_{i_S}\otimes\sigma_{i_T})\bigr)
+=e^{i(q_S+q_T)\theta}\,\tr\bigl(\rho\,\sigma_{i_S}\otimes\sigma_{i_T}\bigr)
+\qquad\text{for all }\theta,
+\end{equation*}
+where $q_S,q_T$ are the total weights of $\sigma_{i_S}$, $\sigma_{i_T}$ in the $\{E_+,E_0,E_-\}$ basis. Hence a source block of total charge $q_S$ couples \emph{only} to a target block of charge $q_T=-q_S$, not $q_T=q_S$: this is the familiar rule that a correlation function survives only between charge-conjugate directions (e.g.\ $\langle S^+S^-\rangle$ need not vanish, $\langle S^+S^+\rangle$ must). For a non-abelian group such as $SU(2)$ this simple picture does \emph{not} extend by using a Cartan generator (e.g.\ $J_z$) in place of the Casimir: diagonalizing $J_z$ only refines each isotypic block further, by magnetic quantum number, and neither reproduces the isotypic block structure of Proposition~\ref{prop:block-diagonal} nor explains why the reduced block $A_\lambda$ of Proposition~\ref{prop:reduced-formula} is independent of that quantum number --- that collapse is a genuine consequence of full non-abelian equivariance, not of commuting with a single conserved charge. The abelian case is special precisely because weight and isotype coincide there.
+\end{remark}
+
+To see this mechanism at work on a state that is not simultaneously covered by the stabilizer mechanism of Lemma~\ref{lem:stabilizer-degeneracy}, consider four qubits $A,B,C,D$ and
+\begin{equation}
+\lvert\psi_\varphi\rangle=\cos\varphi\,\lvert0011\rangle+\sin\varphi\,\lvert1100\rangle,
+\qquad
+\rho_\varphi=\lvert\psi_\varphi\rangle\!\langle\psi_\varphi\rvert,
+\label{eq:u1-example-state}
+\end{equation}
+with $S=\{A,B\}$, $S^c=\{C,D\}$. Both computational-basis terms have Hamming weight $2$, so $\rho_\varphi$ commutes with the collective $U(1)_z$ generated by $R_z(\theta)^{\otimes4}$ for every $\varphi$, but for generic $\varphi$ it is invariant under no larger continuous collective symmetry. A direct computation gives
+\begin{align*}
+\tr(\rho_\varphi\,X^{\otimes4})&=\sin(2\varphi),
+&
+\tr(\rho_\varphi\,X_AX_BY_CY_D)&=-\sin(2\varphi),
+\\
+\tr(\rho_\varphi\,X_AY_BY_CX_D)&=\sin(2\varphi),
+&
+\tr(\rho_\varphi\,Z^{\otimes4})&=1,
+\end{align*}
+with all remaining nonzero-weight correlators vanishing identically; the first three are $\pm1$ only at the isolated points $\varphi=\pi/4\ (\mathrm{mod}\ \pi/2)$, so $\rho_\varphi$ is generically not Pauli-flat and Lemma~\ref{lem:stabilizer-degeneracy} does not apply to it.
+
+For each two-qubit cluster $S$ and $S^c$, the local weight decomposition $\{-1,0,+1\}\otimes\{-1,0,+1\}$ gives total-charge sectors $q=-2,-1,0,1,2$ of dimension $1,2,3,2,1$ respectively (the coefficients of $(x^{-1}+1+x)^2$), on both the source and the target side. By Remark~\ref{rem:abelian-selection-rule} the normalized bigraduated map $\mathcal M_{AB}(\rho_\varphi)$ of Eq.~\eqref{eq:bigraduated-map} block-diagonalizes according to $q_S=-q_T$, and an exact computation of every block gives
+
+\begin{center}
+\begin{tabular}{lccc}
+\toprule
+$q_S\to q_T$ & block size & singular value(s) & origin \\
+\midrule
+$+2\to-2$ & $1\times1$ & $\tfrac23\sin(2\varphi)$ & $E_+E_+\to E_-E_-$ \\
+$+1\to-1$ & $2\times2$ & $0,0$ & identically zero for every $\varphi$ \\
+$0\to0$ & $3\times3$, rank $1$ & $\tfrac13$ & entirely from $E_0E_0\to E_0E_0$, i.e.\ $ZZ\to ZZ$ \\
+$-1\to+1$ & $2\times2$ & $0,0$ & identically zero for every $\varphi$ \\
+$-2\to+2$ & $1\times1$ & $\tfrac23\sin(2\varphi)$ & $E_-E_-\to E_+E_+$ \\
+\bottomrule
+\end{tabular}
+\end{center}
+so that, exactly,
+\begin{equation}
+\norm{\mathcal M_{AB}(\rho_\varphi)}_* = \frac13+\frac43\sin(2\varphi).
+\label{eq:u1-example-formula}
+\end{equation}
+The $q=0$ value $\tfrac13$ is $\varphi$-independent, because $ZZZZ$ is diagonal in the computational basis and both terms of $\lvert\psi_\varphi\rangle$ give it the same eigenvalue $+1$; the entire $\varphi$-dependence, and with it the entanglement signal, sits in the $q=\pm2$ sectors. Equation~\eqref{eq:u1-example-formula} exceeds the cut-separable bound of Theorem~\ref{thm:cluster-cut} exactly when $\sin(2\varphi)>1/2$, i.e.\ for $\varphi\in(\pi/12,\,5\pi/12)$; it vanishes at $\varphi=0$, where $\rho_\varphi$ is itself a product state across $S\mid S^c$, and reaches $\norm{\mathcal M_{AB}(\rho_\varphi)}_*=5/3$ at the maximally superposed point $\varphi=\pi/4$.
+
+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}.
+
+\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 Sections~\ref{sec:multiparty-sources} and~\ref{sec:qubit-numerics}. 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.
+
+\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
+\[
+\rho_H \;=\; \frac{\Pi_H}{\operatorname{rank}\Pi_H}, \qquad \Pi_H=\frac{1}{|H|}\sum_{h\in H} h,
+\]
+the maximally mixed state on the joint $+1$-eigenspace of $H$ (a stabilizer code state; $\rho_H$ is pure iff $|H|=2^n$).
+
+\begin{lemma}[Support of a stabilizer-code state]
+\label{lem:code-support}
+$\operatorname{Tr}[\rho_H\,\sigma_{\vec i}] = \mathbb 1[\vec i\in H]$ for every $\vec i\in\mathbb F_2^{2n}$.
+\end{lemma}
+\begin{proof}
+For $\vec i\in H$: $\Pi_H\sigma_{\vec i}=\Pi_H$ since $h\Pi_H=\Pi_H$ for every $h\in H$, so $\operatorname{Tr}[\Pi_H\sigma_{\vec i}]=\operatorname{Tr}[\Pi_H]=\operatorname{rank}\Pi_H$, giving $\operatorname{Tr}[\rho_H\sigma_{\vec i}]=1$. For $\vec i\notin H$: either $\sigma_{\vec i}$ anticommutes with some $h\in H$, whence $\operatorname{Tr}[\Pi_H\sigma_{\vec i}]=\operatorname{Tr}[h\Pi_H\sigma_{\vec i}]=-\operatorname{Tr}[\Pi_H\sigma_{\vec i}h]=-\operatorname{Tr}[\Pi_H\sigma_{\vec i}]$ (using $h\Pi_H=\Pi_H$ and cyclicity), forcing it to vanish; or $\sigma_{\vec i}$ commutes with all of $H$ without belonging to it (a logical operator), in which case it acts as a nonzero-weight, traceless operator on the logical subspace on which $\rho_H$ restricts to a multiple of the identity, again giving zero.
+\end{proof}
+
+\begin{lemma}[Forced degeneracy of $\widetilde{\mathcal M}_S(\rho_H)$]
+\label{lem:stabilizer-degeneracy}
+Let $S\mid S^c$ be a cut and let $\varphi:H\to\mathbb F_2^{2|S|}$, $\psi:H\to\mathbb F_2^{2|S^c|}$ be the two restriction homomorphisms, so that $H\hookrightarrow \mathbb F_2^{2|S|}\times\mathbb F_2^{2|S^c|}$ via $h\mapsto(\varphi(h),\psi(h))$. If $\psi$ is injective, then every nonzero row of $\widetilde{\mathcal M}_S(\rho_H)$, expressed directly in the (non-orthonormal) raw Pauli-string basis $\{\sigma_{\vec i}\}$ on both sides --- the basis and normalization in which the correlation tensor $C_S(\rho)$ of Eq.~\eqref{eq:corr-tensor-def} and the combined shadow map $\mathcal M_S(\rho)$ of Definition~\ref{def:bigraduated} are actually computed --- has exactly $|\ker\varphi|$ nonzero entries, all of magnitude $1$, with pairwise disjoint column supports across distinct rows; consequently
+\[
+\widetilde{\mathcal M}_S(\rho_H) \text{ has exactly } |\operatorname{im}\varphi|-1 \text{ equal nonzero singular values, each } =\sqrt{|\ker\varphi|}.
+\]
+After the normalization of Eq.~\eqref{eq:bigraduated-map}, the singular values of $\mathcal M_S(\rho_H)$ are therefore
+\[
+\sqrt{\frac{|\ker\varphi|}{(d_S-1)(d_{S^c}-1)}},
+\]
+with multiplicity $|\operatorname{im}\varphi|-1$.
+\end{lemma}
+\begin{proof}
+By Lemma~\ref{lem:code-support}, the entry of $\widetilde{\mathcal M}_S(\rho_H)$ at row $\vec j\in\mathbb F_2^{2|S|}\setminus\{0\}$, column $\vec k$, is $1$ if $(\vec j,\vec k)\in H$ and $0$ otherwise. For fixed $\vec j\in\operatorname{im}\varphi$, the set $\{\vec k:(\vec j,\vec k)\in H\}$ is a coset of $\ker\varphi$ under the group structure of $H$ (standard fiber property of a homomorphism), hence has size $|\ker\varphi|$, giving the row weight and (since all entries are $\pm1$ in magnitude by Lemma~\ref{lem:code-support}) equal row norm $\sqrt{|\ker\varphi|}$ for every nonzero row. If rows for $\vec j\neq\vec j'$ shared a nonzero column $\vec k$, then $(\vec j,\vec k),(\vec j',\vec k)\in H$ would give $(\vec j-\vec j',0)\in H$ with $\vec j\neq\vec j'$, i.e.\ a nontrivial element of $\ker\psi$ --- excluded by injectivity of $\psi$. Rows are thus pairwise orthogonal with equal norm, hence (after normalizing) already the right singular vectors, and the singular values are all equal to the common row norm. The final rescaling is exactly the normalization already applied in Eq.~\eqref{eq:bigraduated-map}.
+\end{proof}
+
+\begin{corollary}
+\label{cor:stabilizer-examples}
+The Lemma applies uniformly to pure stabilizer states ($|H|=2^n$, including all graph states and $\GHZ_n$) and to uniform mixtures over a stabilizer code space with $|H|<2^n$ (including the Smolin state, $H=\{IIII,XXXX,YYYY,ZZZZ\}$). Injectivity of $\psi$ holds automatically whenever $H$ contains no element supported entirely on $S$ --- checkable by inspection of the generators, without any Lie-group input.
+
+For $\GHZ_3$ with $S=\{1\}$: $H=\{III,XXX,ZZI,ZIZ,IZZ,YYX,YXY,XYY\}$ (up to signs), $|\ker\varphi|=2$, $|\operatorname{im}\varphi|=4$, $(d_S-1)(d_{S^c}-1)=1\times3=3$, giving $3$ singular values equal to $\sqrt{2/3}$ --- exactly the value reported in Section~\ref{sec:qubit-numerics}.
+
+For the Smolin state, every $1\mid3$ and $2\mid2$ cut has $\varphi$ bijective ($|\ker\varphi|=1$), so $|\operatorname{im}\varphi|-1=3$ in both cases --- bijectivity of $\varphi$ holds for \emph{any} nonempty proper subset $S$ of the four legs given this particular $H$. This gives $3$ equal singular values throughout, with value
+\[
+\sqrt{\frac{1}{(d_S-1)(d_{S^c}-1)}}
+=
+\begin{cases}
+1/\sqrt7 & \text{for the } 1\mid3 \text{ cut } ((d_S-1)(d_{S^c}-1)=1\times7),\\[2pt]
+1/3 & \text{for the } 2\mid2 \text{ cut } ((d_S-1)(d_{S^c}-1)=3\times3),
+\end{cases}
+\]
+matching Section~\ref{sec:multiparty-sources} and Section~\ref{sec:qubit-numerics} exactly.
+\end{corollary}
+
+\begin{remark}
+The $y$-parity grading occasionally useful for real-in-the-computational-basis states is the special case $H=\{I^{\otimes n}\}$ acting trivially --- more precisely, it is not itself an instance of this Lemma but a compatible, coarser $\mathbb Z_2$-grading that commutes with any $H$-decomposition and can be applied on top of it without modification.
+\end{remark}
+
+\begin{remark}[Two complementary mechanisms]
+\label{rem:two-mechanisms}
+Proposition~\ref{prop:block-diagonal} and Lemma~\ref{lem:stabilizer-degeneracy} are complementary, not competing, and apply under disjoint hypotheses. The representation-theoretic mechanism applies whenever $\rho$ is genuinely $G$-invariant under some compact group $G$ acting by local unitaries and preserving the cut, regardless of whether $\rho$ is Pauli-diagonal; it says nothing about states, such as a generic finite-group-symmetric state built from a permutation representation, that are not Pauli-diagonal. The stabilizer mechanism applies whenever $\rho$ is (a uniform mixture over) a stabilizer code state, regardless of whether it possesses any continuous symmetry at all --- as is the case for $\GHZ_n$, the Smolin state, and every graph state used elsewhere in this note, none of which is invariant under a nontrivial continuous collective symmetry. In the (comparatively narrow) overlap where a state is both $G$-invariant for some continuous $G$ and Pauli-diagonal, both mechanisms apply and constrain the same block structure from different directions; outside that overlap, exactly one of the two is available, and it is this Lemma, not Proposition~\ref{prop:block-diagonal}, that accounts for every numerically observed degeneracy reported so far in this note.
+\end{remark}
+
 \section{Outlook}
 
 The combined shadow map should be viewed as a structured refinement of the older correlation strengths $L_S$ introduced by Aschauer \emph{et al.} \cite{aschauer}. Those quantities keep one Frobenius norm per tensor block; the present construction keeps the common one-vs-rest channel structure across all orthogonal sectors on the complement. In that sense it preserves the geometric spirit of that local-invariant sector decomposition while extracting more information from the same correlation data.