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\title{Symmetry-adapted block decomposition of the shadow map}
\author{Draft formal note}
\date{June 2026}

\begin{document}

\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}
The bigraduated shadow map $\mathcal M_S(\rho)$ of the companion note
\cite{aschauer2026a} recovers known cut-separable bounds while exposing a
much finer block structure than a single collapsed correlation matrix.
Several features of that structure --- the triple degeneracy of
$\mathcal M_a(\GHZ_3)$'s singular values, the analogous triple
degeneracy of the Smolin state under both the $1\mid3$ and $2\mid2$ cut,
and the common saturating value $6/\sqrt7$ shared by all $38$ connected
four-qubit graph states --- were reported there as separate numerical
facts. We show these degeneracies are forced \emph{exactly}, by two
structurally independent mechanisms. The first is representation-theoretic:
whenever $\rho$ is invariant under a compact group acting by unitaries
compatible with the cut, Schur's lemma forces $\mathcal M_S(\rho)$ to be
block diagonal over isotypic components; we give the precise (weak,
cut-factorizing versus strong, party-local) hypotheses under which this
is compatible with the existing sector grading, and exhibit an axial-$U(1)$
four-qubit example, not itself a stabilizer state, where this mechanism
alone is responsible. The second is combinatorial: for any Pauli-diagonal
(stabilizer-code) state, an elementary $\mathbb F_2$-linear argument on
the restriction of the stabilizer group to each side of the cut fixes the
multiplicity and common value of a degenerate block exactly, with no
reference to continuous symmetry, recovering in closed form every
previously numerical GHZ$_3$-, Smolin-, and graph-state degeneracy of
\cite{aschauer2026a}. Building on this degeneracy structure, we develop
first-order degenerate perturbation theory for the shadow map's singular
values via a Jordan--Wielandt dilation, and show that single-qubit Pauli
dephasing rescales the shadow map \emph{exactly}, to all orders, along
axes fixed by commutation with the stabilizer; worked out completely for
$\GHZ_3$, this isolates a correlation channel exactly protected against
one dephasing axis while decaying twice as fast along the other two.
Finally, we show that the same isotypic collapse becomes a genuine
computational tool once a cluster is invariant under the \emph{full}
permutation group of its constituent parties: the corresponding
shadow-map block reduces, without approximation, from a $3^m\times3^l$
matrix to one of polynomial size $\binom{m+2}2\times\binom{l+2}2$, which
we demonstrate for Dicke states with cluster sizes up to $m=40$, $l=35$
within an $n=200$-qubit state.
\end{abstract}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}
\label{sec:introduction}

The companion note~\cite{aschauer2026a} introduces the bigraduated
shadow map $\mathcal M_S(\rho)$: for a source cluster $S$ and its
complement $S^c$, this stacks every sector-restricted correlation-response
block $M_{V\to T}(\rho)$, $V\subseteq S$, $T\subseteq S^c$, into a single
object whose nuclear norm is bounded by $1$ on states separable across
$S\mid S^c$ (Theorem~\ref{thm:cluster-cut}, recalled in
Section~\ref{sec:setup}). Working through examples there repeatedly
turned up the same phenomenon: at highly symmetric states --- $\GHZ_3$,
the Smolin state, every connected four-qubit graph state --- the singular
values of $\mathcal M_S(\rho)$ come out not merely close to, but
\emph{exactly}, equal. In~\cite{aschauer2026a} this was recorded as a
numerical fact at each example separately. This note asks, and answers,
the obvious next question: what forces this, and is it the same
mechanism every time?

The refinement this note develops is worth having for four distinct
reasons, which we state before the formal statements, since they are not
equally strong and motivate different parts of what follows.
\begin{itemize}
\item \emph{Explanatory power.} The degeneracies just listed convert from
separately verified numerical observations into structural statements,
proved once and for all.
\item \emph{Diagnostic resolution.} The sub-block witnesses of
Corollary~\ref{cor:sub-block} resolve a bound violation only down to
\emph{which parties} are involved. A representation-theoretic
decomposition, where applicable, resolves it further, down to \emph{which
symmetry channel} within a fixed sector is responsible.
\item \emph{Computational cost.} An isotypic decomposition replaces one
singular value decomposition on the full sector space by several
independent, much smaller ones on the multiplicity spaces, whose
dimensions grow far more slowly than the ambient sector dimension.
\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 $\rho$,
so a block-diagonal criterion can serve as an inexpensive pre-test.
\end{itemize}

We show the answer to the motivating question is genuinely twofold, and
that the two mechanisms are logically independent rather than special
cases of one another (Remark~\ref{rem:two-mechanisms}):
\begin{itemize}
\item[(i)] a \emph{representation-theoretic} mechanism
(Section~\ref{sec:mechanisms}, first part): whenever $\rho$ is invariant
under a compact group $G$ acting by unitaries compatible with the cut,
$\mathcal M_S(\rho)$ block-diagonalizes exactly over the isotypic
decomposition of $\V_0^{(S)}$, $\V_0^{(S^c)}$ under $G$
(Proposition~\ref{prop:block-diagonal}); under the stronger, party-local
form of the hypothesis, this block decomposition is moreover compatible
with, and refines, the combinatorial sector grading already used to
define $\mathcal M_S$ (Proposition~\ref{prop:sector-symmetry-compatible},
Corollary~\ref{cor:joint-refinement});
\item[(ii)] a \emph{combinatorial, stabilizer-support} mechanism
(Section~\ref{sec:mechanisms}, second part): for any Pauli-diagonal
state --- a uniform mixture on the joint eigenspace of an isotropic
subgroup $H\le\mathbb F_2^{2n}$, including every stabilizer state and
every graph state --- the multiplicity and common value of a degenerate
singular-value block are fixed exactly by an elementary counting argument
on the restriction homomorphisms of $H$ to each side of the cut
(Lemma~\ref{lem:stabilizer-degeneracy}), with no reference to any
continuous symmetry at all.
\end{itemize}
$\GHZ_3$, the Smolin state, and the four-qubit graph-state family are all
Pauli-diagonal but carry no useful continuous collective symmetry, so
mechanism~(ii) alone explains their degeneracies
(Corollary~\ref{cor:stabilizer-examples}); conversely, the
axial-$U(1)$-symmetric four-qubit example of Section~\ref{sec:mechanisms}
is generically not Pauli-diagonal, so only mechanism~(i) is available
there. The two mechanisms overlap only for states that are both
symmetric and stabilizer, where they constrain the same block from two
independent directions.

Having identified why the shadow map degenerates so often, two further
questions become natural, and Sections~\ref{sec:perturbation} and
\ref{sec:polynomial-witness} answer them.

\emph{What happens under a generic perturbation away from a symmetric
point?} Since $\rho\mapsto\mathcal M_S(\rho)$ is exactly linear, all of
the interesting structure sits in the nuclear norm, which is only
directionally differentiable at a point of repeated singular value
\cite{lewissendov2005}. Section~\ref{sec:perturbation} develops the
relevant first-order degenerate perturbation theory via a
Jordan--Wielandt dilation (Theorem~\ref{thm:degenerate-svd-perturbation}),
identifies a single real symmetric compression matrix governing the
splitting, and specializes this completely to single-qubit Pauli
dephasing on $\GHZ_3$: dephasing along the stabilizer's own Pauli type
turns out to rescale the shadow map \emph{exactly}, not just to leading
order (Proposition~\ref{prop:pauli-dephasing-exact}), isolating one
correlation channel that survives a whole one-parameter family of noise
unchanged (Corollary~\ref{cor:ghz3-exact-channel}) while the other two
decay at twice the generic rate.

\emph{Can the same symmetry that forces degeneracy also be exploited
computationally?} The bigraduation itself costs up to $O(2^n)$
combinatorial blocks in the worst case (Section~6 of
\cite{aschauer2026a}). Section~\ref{sec:polynomial-witness} shows this
obstruction disappears when a cluster is invariant under the \emph{full}
permutation group of its own parties: the relevant shadow-map block
collapses, exactly and without approximation, from a $3^m\times3^l$
matrix to one of size $\binom{m+2}2\times\binom{l+2}2$, polynomial rather
than exponential in cluster size (Proposition~\ref{prop:multinomial-collapse}),
which we use to evaluate the corresponding witness for Dicke states with
clusters of up to $m=40$, $l=35$ qubits within an $n=200$-qubit state
(Table~\ref{tab:dicke-scaling}), far beyond what the ambient matrix could
represent explicitly.

Section~\ref{sec:setup} fixes notation and recalls, without reproof, the
minimal apparatus from~\cite{aschauer2026a} needed below: the sector
decomposition of $\V_0^{(S)}$, $\V_0^{(S^c)}$, the definition of
$\mathcal M_S(\rho)$, and the cut-separable bound. Sections
\ref{sec:mechanisms}--\ref{sec:polynomial-witness} develop the results
just previewed, and Section~\ref{sec:outlook} closes with the questions
this raises in turn.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Setup and recap}
\label{sec:setup}

This section fixes notation and recalls, without reproof, the minimal
apparatus from the companion note~\cite{aschauer2026a} used below: the
generator convention, the sector decomposition of the local operator
spaces, the definition of the bigraduated shadow map $\mathcal M_S(\rho)$,
and the cut-separable bound it satisfies. Readers already familiar with
\cite{aschauer2026a} can skip ahead to Section~\ref{sec:mechanisms}; full
derivations, the tensor-unfolding viewpoint, sub-block witnesses beyond
Corollary~\ref{cor:sub-block}, and the qubit/qutrit numerics referenced
informally throughout this note are all developed there.

\subsection{Parties, generators, and the qubit convention}
\label{sec:qubit-convention}

Let $P=\{1,\dots,n\}$ be a set of parties with local Hilbert spaces
$\mathcal H^{(a)}$ of dimension $d_a$. For each party $a$, fix traceless
Hermitian generators $\sigma_1^{(a)},\dots,\sigma_{d_a^2-1}^{(a)}$ together
with the identity $\sigma_0^{(a)}=\id$, normalized by
\begin{equation}
\tr\bigl(\sigma_i^{(a)}\sigma_j^{(a)}\bigr)=d_a\,\delta_{ij},
\qquad i,j\in\{0,1,\dots,d_a^2-1\},
\label{eq:generator-orthogonality}
\end{equation}
following the local-invariant sector decomposition of Aschauer
\emph{et al.}~\cite{aschauer}. For nonempty $R\subseteq P$, the $R$-body
correlation tensor of $\rho$ is
\begin{equation}
C_R(\rho)=\bigl(c_{i_a:a\in R}\bigr),
\qquad
c_{i_a:a\in R}=\tr\Bigl(\rho\bigotimes_{a\in R}\sigma_{i_a}^{(a)}\Bigr),
\quad 1\le i_a\le d_a^2-1.
\label{eq:corr-tensor-def}
\end{equation}

For qubits ($d_a=2$ for every $a\in P$), the setting of every example in
this note, we fix once and for all
\begin{equation*}
\sigma_0=\id,\qquad \sigma_1=X,\qquad \sigma_2=Y,\qquad \sigma_3=Z.
\end{equation*}
(For the qutrit Tiles benchmark of \cite{aschauer2026a}, not otherwise
needed here, the generators are instead Gell-Mann matrices rescaled by
$\sqrt{3/2}$, so that \eqref{eq:generator-orthogonality} holds with
$d_a=3$.)

\subsection{Sector decomposition}
\label{sec:sector-decomp-recap}

Let $\V^{(b)}$ be the real Hilbert space of Hermitian operators on
$\mathcal H^{(b)}$, equipped with the Hilbert--Schmidt inner product, and
let $\V_0^{(b)}:=\vrspan\{\sigma_1^{(b)},\dots,\sigma_{d_b^2-1}^{(b)}\}$ be
the traceless local subspace. For any nonempty $R\subseteq P$, the
operator space $\V^{(R)}:=\bigotimes_{b\in R}\V^{(b)}$ decomposes
orthogonally as
\begin{equation}
\V^{(R)}=\V_\emptyset^{(R)}\oplus\bigoplus_{\emptyset\neq V\subseteq R}^\perp\V_V^{(R)},
\qquad
\V_V^{(R)}:=\vrspan\Bigl\{\textstyle\bigotimes_{b\in R}\sigma_{i_b}^{(b)}\ :\ i_b\neq0\iff b\in V\Bigr\},
\label{eq:source-bloch-decomposition}
\end{equation}
with $\V_\emptyset^{(R)}=\vrspan\{\id_R\}$. We use
\eqref{eq:source-bloch-decomposition} both on the source side, $R=S$, for
a fixed nonempty $S\subsetneq P$, and on the target side, $R=S^c:=P\setminus S$;
the corresponding sector projectors $P_V:\V^{(S)}\to\V_V^{(S)}$, $V\subseteq
S$, are exactly those referenced in Proposition~\ref{prop:sector-symmetry-compatible}
below.

\subsection{The bigraduated shadow map}
\label{sec:bigraduated-recap}

The intrinsic source-to-complement response operator
$\widetilde{\mathcal M}_S(\rho):\V_0^{(S)}\to\V_0^{(S^c)}$ is defined by
\begin{equation*}
\langle Y,\widetilde{\mathcal M}_S(\rho)X\rangle=\tr\bigl(\rho(X\otimes Y)\bigr),
\qquad X\in\V_0^{(S)},\ Y\in\V_0^{(S^c)}.
\end{equation*}
Writing $\iota_V:\V_V^{(S)}\hookrightarrow\V_0^{(S)}$ for the inclusion of
a source sector and $P_T:\V_0^{(S^c)}\to\V_T^{(S^c)}$ for the projection
onto a target sector, both from \eqref{eq:source-bloch-decomposition}, set
\begin{equation*}
M_{V\to T}(\rho):=P_T\,\widetilde{\mathcal M}_S(\rho)\,\iota_V,
\qquad \emptyset\neq V\subseteq S,\ \emptyset\neq T\subseteq S^c.
\end{equation*}

\begin{definition}[Bigraduated shadow map]
\label{def:bigraduated}
With $d_S:=\prod_{a\in S}d_a$ and $d_{S^c}:=\prod_{a\in S^c}d_a$, the
\emph{combined bigraduated shadow map} is
\begin{equation}
\mathcal M_S(\rho):=\frac{1}{\sqrt{(d_S-1)(d_{S^c}-1)}}\bigoplus_{\substack{\emptyset\neq V\subseteq S\\ \emptyset\neq T\subseteq S^c}}M_{V\to T}(\rho),
\label{eq:bigraduated-map}
\end{equation}
i.e.\ the matrix of $\widetilde{\mathcal M}_S(\rho)$ in sector-adapted
orthonormal coordinates on both sides. For a singleton source $S=\{a\}$,
the source-side decomposition~\eqref{eq:source-bloch-decomposition} has
only the summand $V=S$, and \eqref{eq:bigraduated-map} reduces to the
ordinary, target-only combined shadow map $\mathcal M_a(\rho)$ used
throughout the qubit examples below.
\end{definition}

\subsection{The cut-separable bound}
\label{sec:cut-bound-recap}

\begin{theorem}[Cluster cut-separable bound]
\label{thm:cluster-cut}
If $\rho$ is separable across the cut $S\mid S^c$, then
$\norm{\mathcal M_S(\rho)}_*\le1$. Consequently,
$\norm{\mathcal M_S(\rho)}_*>1$ certifies that $\rho$ is entangled across
$S\mid S^c$.
\end{theorem}

The proof, given in full in \cite{aschauer2026a}, reduces to the product
case $\rho=\rho_S\otimes\sigma_{S^c}$: there
$\widetilde{\mathcal M}_S(\rho)=v_{S^c}(r^{(S)})^T$ is rank one, with
$r^{(S)}$, $v_{S^c}$ the traceless Bloch vectors of $\rho_S$,
$\sigma_{S^c}$, and the correlation-sum identity
$\norm{r^{(S)}}^2=d_S\tr(\rho_S^2)-1\le d_S-1$ (and its target-side
analogue) bounds each factor. Convexity of the nuclear norm extends the
bound from product states to arbitrary mixtures separable across the cut.

\begin{corollary}[Sub-block witnesses]
\label{cor:sub-block}
For any nonempty families $\mathcal V\subseteq\{V:\emptyset\neq V\subseteq S\}$,
$\mathcal T\subseteq\{T:\emptyset\neq T\subseteq S^c\}$, let
$P_{\mathcal V}$, $P_{\mathcal T}$ be the corresponding orthogonal
projections. If $\rho$ is separable across $S\mid S^c$, then
\begin{equation*}
\norm{P_{\mathcal T}\,\mathcal M_S(\rho)\,P_{\mathcal V}}_*\ \le\ \norm{\mathcal M_S(\rho)}_*\ \le\ 1,
\end{equation*}
with no separate proof required: orthogonal projections are contractive
for the operator norm.
\end{corollary}

Everything in Sections~\ref{sec:mechanisms}--\ref{sec:polynomial-witness}
below is built from Definition~\ref{def:bigraduated} and
Theorem~\ref{thm:cluster-cut} alone; no further result of
\cite{aschauer2026a} is used as a technical ingredient, though several are
referenced informally as motivating context.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Two mechanisms of exact degeneracy}
\label{sec:mechanisms}

The two results below need genuinely different amounts of symmetry, and it is worth separating them explicitly rather than proving both under the stronger hypothesis throughout.

\emph{Weak (cut-factorizing) hypothesis.} Let $G$ be a compact group acting on the system by unitaries of the form $g\mapsto U_g^{(S)}\otimes U_g^{(S^c)}$ for some fixed bipartition $S\mid S^c$ -- i.e.\ $G$ acts unitarily on each side of the cut, with no further requirement on how $U_g^{(S)}$ itself decomposes over the parties inside $S$ -- and suppose $\rho$ is $G$-invariant. This induces adjoint representations $\mathrm{Ad}^{(S)}_g$ on $\V_0^{(S)}$ and $\mathrm{Ad}^{(S^c)}_g$ on $\V_0^{(S^c)}$, with isotypic decompositions
\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 \cite{aschauer2026a}, now taken with respect to a representation-theoretic rather than a combinatorial grading; the two coincide only in special cases.

This weak hypothesis is exactly what Proposition~\ref{prop:block-diagonal} below uses: its proof only invokes equivariance of $\widetilde{\mathcal M}_S(\rho)$ between the two representations $\mathrm{Ad}^{(S)}_g$, $\mathrm{Ad}^{(S^c)}_g$, and never asks how $U_g^{(S)}$ or $U_g^{(S^c)}$ further factor over the parties within $S$ or within $S^c$.

\emph{Strong (party-local) hypothesis.} Proposition~\ref{prop:sector-symmetry-compatible} and Corollary~\ref{cor:joint-refinement} need more: there $G$ must act by unitaries of the form $g\mapsto\bigotimes_{a\in P}U_g^{(a)}$, one factor per party, since the sector grading $V\subseteq S$ that these results are compatible with is itself defined party by party (Eq.~\eqref{eq:source-bloch-decomposition}). Every party-local action is in particular cut-factorizing for any bipartition, so the strong hypothesis is a special case of the weak one; the converse fails, as the remark following Corollary~\ref{cor:joint-refinement} makes concrete.

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 weak, cut-factorizing hypothesis 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 \cite{aschauer2026a}, $\V_0^{(S)}=\bigoplus_{\emptyset\neq V\subseteq S}\V_V^{(S)}$. 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 under the strong, party-local hypothesis, $g\mapsto\bigotimes_{a\in P}U_g^{(a)}$, 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 \cite{aschauer2026a} 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}

\begin{remark}[Party permutation: an example separating the two hypotheses]
\label{rem:permutation-separating-example}
Let $\pi\in S_n$ be a permutation of the parties fixing $S$ setwise (i.e.\ $\pi(S)=S$, hence also $\pi(S^c)=S^c$), and let $U_\pi$ be the corresponding permutation unitary on $\mathcal H^{(P)}$. If $\rho$ is invariant under $U_\pi$, this is a cut-factorizing symmetry -- it acts as some unitary on $\mathcal H^{(S)}$ tensored with some unitary on $\mathcal H^{(S^c)}$ -- so Proposition~\ref{prop:block-diagonal} applies unchanged and $\widetilde{\mathcal M}_S(\rho)$ block-diagonalizes over the isotypic components of the induced permutation representations on $\V_0^{(S)}$, $\V_0^{(S^c)}$. But $U_\pi$ is not party-local whenever $\pi$ moves two or more parties within $S$: it acts as a genuine swap between the factors of $\mathcal H^{(S)}$, not as $\bigotimes_{a\in S}U_g^{(a)}$ for a single fixed single-party unitary $U_g$. Consequently Proposition~\ref{prop:sector-symmetry-compatible} does not apply, and indeed its conclusion can fail: $U_\pi$ can map a sector $\V_V^{(S)}$ to a different sector $\V_{\pi(V)}^{(S)}$ rather than fixing it, so the combinatorial sector grading and the permutation-isotypic grading are, in general, genuinely different, incompatible refinements of $\V_0^{(S)}$ -- unlike the party-local case of Corollary~\ref{cor:joint-refinement}, where the two gradings were shown to be compatible refinements of one another.

Two families used elsewhere in this note calibrate this gap. The state $W_n$ is invariant under the full party-permutation group $S_n$, and permutation-invariant under any $\pi$ fixing any given $S$ setwise; it therefore block-diagonalizes under Proposition~\ref{prop:block-diagonal} for every cut, yet it is not a stabilizer state (its single-party marginal has eigenvalues $1/n,(n-1)/n$, never the flat spectrum that Lemma~\ref{lem:code-support} forces on a stabilizer-code reduction), so its degeneracies are explained by permutation symmetry alone, with neither Proposition~\ref{prop:sector-symmetry-compatible} nor Lemma~\ref{lem:stabilizer-degeneracy} available. The ring graph state, by contrast, is invariant only under the cyclic subgroup $C_n\le S_n$, which for $n=4$ and $S=\{1,2\}$ already fails to fix $S$ setwise (the generating shift sends $\{1,2\}\mapsto\{2,3\}$), so Proposition~\ref{prop:block-diagonal} does not apply to that cut at all in the permutation channel; its observed degeneracies, as Corollary~\ref{cor:stabilizer-examples} already notes, are accounted for entirely by the stabilizer mechanism instead. Party-permutation symmetry is thus a third selection mechanism, logically independent of both the collective-unitary mechanism of Proposition~\ref{prop:block-diagonal} under the strong hypothesis and the stabilizer mechanism of Lemma~\ref{lem:stabilizer-degeneracy}, and satisfying it need not entail satisfying either of the other two.
\end{remark}

\begin{example}[Ring graph state: reflections, not rotations, supply the example]
\label{ex:ring-reflection}
The cyclic group $C_4=\langle(1234)\rangle$ stabilizing the ring graph state $\rho_\square$ of \cite{aschauer2026a} indeed fixes no nontrivial cut setwise, as anticipated above. But $\mathrm{Aut}(\square)$ is the full dihedral group $D_4$, of order $8$, and its reflections do fix cuts setwise. For $S=\{1,2\}$, $S^c=\{3,4\}$ (the ``adjacent'' cut of \cite{aschauer2026a}), the reflection $\pi=(12)(34)$ is a graph automorphism with $\pi(S)=S$, $\pi(S^c)=S^c$; it is a genuine two-party swap on each side of the cut, hence exactly the situation of Remark~\ref{rem:permutation-separating-example}: the weak hypothesis holds for $\rho_\square$ at this cut, the strong one does not, since swapping qubits $1$ and $2$ is not of the form $U^{(1)}\otimes U^{(2)}$.

Concretely, $R(\pi)$ does not fix $\V_{\{1\}}^{(S)}$: it maps it onto $\V_{\{2\}}^{(S)}$, since $\pi$ relabels qubit $1$ as qubit $2$. So Proposition~\ref{prop:sector-symmetry-compatible}'s hypothesis fails non-vacuously here, exactly as claimed. What Proposition~\ref{prop:block-diagonal} (the weak statement) still guarantees is equivariance of the full block $\widetilde{\mathcal M}_S(\rho_\square)$ under the combined swap on $\V_{\{1\}}^{(S)}\oplus\V_{\{2\}}^{(S)}$ and $\V_{\{3\}}^{(S^c)}\oplus\V_{\{4\}}^{(S^c)}$; restricted to the four singleton-to-singleton blocks this is the concrete prediction
\begin{equation}
M_{\{1\}\to\{3\}}(\rho_\square)=M_{\{2\}\to\{4\}}(\rho_\square),
\qquad
M_{\{1\}\to\{4\}}(\rho_\square)=M_{\{2\}\to\{3\}}(\rho_\square),
\label{eq:ring-circulant-prediction}
\end{equation}
i.e.\ that the $2\times2$ array of blocks is circulant --- a prediction that follows from equivariance alone, before any stabilizer computation is performed.

A direct check against the stabilizer group $H=\langle K_1,K_2,K_3,K_4\rangle$ of $\rho_\square$ (Lemma~\ref{lem:code-support}) confirms this: in the raw Pauli-string basis, $X_1X_3\in H$ and $X_2X_4\in H$, while $X_1X_4\notin H$ and $X_2X_3\notin H$, so
\begin{equation*}
M_{\{1\}\to\{3\}}(\rho_\square)=M_{\{2\}\to\{4\}}(\rho_\square)=1\ (\text{support on the }xx\text{ direction only}),
\end{equation*}
\begin{equation*}
M_{\{1\}\to\{4\}}(\rho_\square)=M_{\{2\}\to\{3\}}(\rho_\square)=0,
\end{equation*}
matching Eq.~\eqref{eq:ring-circulant-prediction} exactly. This is a case where the weak, cut-factorizing symmetry and the combinatorial stabilizer mechanism of Lemma~\ref{lem:stabilizer-degeneracy} operate on the same numbers from two independent directions: the stabilizer mechanism explains \emph{why} the entries are $0$ or $\pm1$ at all, while the swap-equivariance of Proposition~\ref{prop:block-diagonal} explains, without reference to the stabilizer formalism, why the two nonzero entries must coincide and the two zero entries must coincide. Diagonalizing this $2\times2$ circulant block in the symmetric/antisymmetric basis $(\,e_x^{(1)}\pm e_x^{(2)})/\sqrt2$ gives eigenvalues $1\pm0=1$ in this instance --- here the two channels happen to be degenerate, since the off-diagonal entry vanishes, but the circulant form Eq.~\eqref{eq:ring-circulant-prediction} would hold with generically distinct symmetric/antisymmetric eigenvalues $a\pm b$ for any $\rho$ merely invariant under this same reflection, stabilizer or not.

The ``diagonal'' cut $S=\{1,3\}$, $S^c=\{2,4\}$ of \cite{aschauer2026a} admits the analogous reflection $\pi'=(13)$ (fixing qubits $2,4$ individually and swapping $1,3$ within $S$), giving the same story with $S$ and $S^c$ playing asymmetric roles: here $\pi'$ swaps only within $S$, not within $S^c$, so the weak hypothesis still applies, but now with $R_{S^c}(\pi')$ trivial and $R_S(\pi')$ a nontrivial swap. Equivariance then forces the weaker, one-sided prediction $M_{\{1\}\to T}(\rho_\square)=M_{\{3\}\to T}(\rho_\square)$ for every $T\subseteq\{2,4\}$, again checkable directly against $H$.
\end{example}

\begin{lemma}[Transitive party symmetry forces cut-value equality]
\label{lem:transitive-symmetry}
Suppose all local dimensions coincide, $d_a=d$ for every $a\in P$, and let $H\le S_n$ act on $P$ by permuting parties via the induced permutation unitaries $U_\pi$ on $\mathcal H^{(P)}=(\mathcal H^{(1)})^{\otimes n}$. Suppose $\rho$ is $H$-invariant, $U_\pi\rho U_\pi^\dagger=\rho$ for every $\pi\in H$. Then for every $a\in P$ and $\pi\in H$,
\begin{equation}
\norm{\mathcal M_{\pi(a)}(\rho)}_*=\norm{\mathcal M_a(\rho)}_*.
\label{eq:transitive-equal-norms}
\end{equation}
Consequently, if $H$ acts transitively on $P$, the value $\norm{\mathcal M_a(\rho)}_*$ is independent of $a$, and
\begin{equation}
\Phi_{\mathrm{sym}}(\rho)=\Phi_{\max}(\rho)=\norm{\mathcal M_a(\rho)}_*
\label{eq:transitive-collapse}
\end{equation}
for any single choice of $a\in P$ -- in particular, computing one shadow map suffices instead of all $n$.
\end{lemma}

\begin{proof}
Fix $\pi\in H$ and $a\in P$. Relabeling parties by $\pi$ is a relabeling of tensor factors, so $\mathcal M_{\pi(a)}(U_\pi\rho U_\pi^\dagger)$ is, by construction, exactly the same operator as $\mathcal M_a(\rho)$ transported through the canonical identification of $\mathcal H^{(a)}$ with $\mathcal H^{(\pi(a))}$ (which exists since $d_a=d$ for all $a$); in particular it has the same singular values, so
\begin{equation*}
\norm{\mathcal M_{\pi(a)}\bigl(U_\pi\rho U_\pi^\dagger\bigr)}_*=\norm{\mathcal M_a(\rho)}_*.
\end{equation*}
By hypothesis $U_\pi\rho U_\pi^\dagger=\rho$, giving Eq.~\eqref{eq:transitive-equal-norms}. If $H$ is transitive, the orbit of any fixed $a$ under $H$ is all of $P$, so $\norm{\mathcal M_b(\rho)}_*$ takes the same value for every $b\in P$; Eq.~\eqref{eq:transitive-collapse} is then immediate from the definitions of $\Phi_{\mathrm{sym}}$, $\Phi_{\max}$.
\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 the cut bound in \cite{aschauer2026a}, 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 \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.

\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 \cite{aschauer2026a}.

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 the results in \cite{aschauer2026a} 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}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55


\section{Degenerate perturbation theory for the shadow map}
\label{sec:perturbation}

The states used throughout this note to calibrate the degeneracy
mechanisms of Proposition~\ref{prop:block-diagonal} and
Lemma~\ref{lem:stabilizer-degeneracy} --- $\GHZ_3$, the Smolin state,
the four-qubit graph states --- are all highly symmetric points at
which the shadow map has repeated singular values. A natural next
question is how these degeneracies behave under a generic perturbation
away from the symmetric point, e.g.\ under physically motivated local
noise. Since $\rho\mapsto\mathcal M_S(\rho)$ is exactly linear
(Eq.~\eqref{eq:bigraduated-map}), the map itself carries no new
structure under $\rho(\varepsilon)=\rho_0+\varepsilon\,\delta\rho$: one
has $\mathcal M_S(\rho(\varepsilon))=\mathcal M_S(\rho_0)+\varepsilon\,
\mathcal M_S(\delta\rho)$ identically, for every $\varepsilon$. All of
the interesting structure is instead in the \emph{nuclear norm}
$\norm{\mathcal M_S(\rho(\varepsilon))}_*$, which is only convex, not
linear, and is generically non-differentiable exactly at points of
repeated singular value \cite{lewissendov2005}. We now make this
precise and then work it out completely for $\GHZ_3$.

\begin{theorem}[First-order splitting of degenerate shadow singular values]
\label{thm:degenerate-svd-perturbation}
Let $M:=\mathcal M_S(\rho_0)\in\R^{N\times n}$ have an $m$-fold
degenerate singular value $\sigma_0>0$, with orthonormal singular
bases $U_0\in\R^{N\times m}$, $V_0\in\R^{n\times m}$ satisfying
$MV_0=\sigma_0 U_0$, $M^TU_0=\sigma_0V_0$. For a perturbation direction
$\delta\rho$ with $\rho(\varepsilon):=\rho_0+\varepsilon\,\delta\rho$ a
valid state for $\varepsilon$ in some interval around $0$, define the
symmetric compression
\begin{equation}
K:=\frac12\Bigl(U_0^T\,\mathcal M_S(\delta\rho)\,V_0
+V_0^T\,\mathcal M_S(\delta\rho)^T\,U_0\Bigr)\in\R^{m\times m}.
\label{eq:K-matrix-def}
\end{equation}
Then the $m$ singular values of $\mathcal M_S(\rho(\varepsilon))$
branching from $\sigma_0$ satisfy
\begin{equation}
\sigma_i(\varepsilon)=\sigma_0+\varepsilon\,\lambda_i(K)+O(\varepsilon^2),
\qquad i=1,\dots,m,
\label{eq:svd-splitting}
\end{equation}
where $\lambda_i(K)$ are the (real) eigenvalues of $K$.
\end{theorem}

\begin{proof}
Embed $M$ into the symmetric Jordan--Wielandt dilation
$H(M):=\begin{pmatrix}0&M\\M^T&0\end{pmatrix}\in\R^{(N+n)\times(N+n)}$.
If $M=\sum_k\sigma_k u_kv_k^T$ is a singular value decomposition, then
$H(M)$ has eigenpairs $\bigl(\pm\sigma_k,\,\tfrac1{\sqrt2}(u_k,\pm v_k)\bigr)$
for every $k$, together with a zero eigenspace filling out the
orthogonal complements on each side; this is a standard fact (see
e.g.\ \cite{kato1995perturbation}). Since $\rho\mapsto\mathcal M_S(\rho)$
is linear, $H(M(\rho(\varepsilon)))=H(M(\rho_0))+\varepsilon\,H(M(\delta\rho))$
exactly, i.e.\ $H$ depends linearly on $\varepsilon$. The eigenvalue
$\sigma_0$ of the real symmetric matrix $H(M(\rho_0))$ has eigenspace
spanned by the orthonormal vectors $w_k:=\tfrac1{\sqrt2}(u_{0,k},v_{0,k})$,
$k=1,\dots,m$ (columns of $U_0$, $V_0$). Ordinary degenerate
Rayleigh--Schr\"odinger perturbation theory for symmetric matrices
\cite{kato1995perturbation} then gives that the $m$ eigenvalues of
$H(M(\rho(\varepsilon)))$ branching from $\sigma_0$ equal
$\sigma_0+\varepsilon\,\lambda_i(W^TH(M(\delta\rho))W)+O(\varepsilon^2)$,
where $W=(w_1,\dots,w_m)$. A direct computation gives
$(W^TH(M(\delta\rho))W)_{kl}=\tfrac12\bigl(u_{0,k}^T\mathcal M_S(\delta\rho)v_{0,l}
+v_{0,k}^T\mathcal M_S(\delta\rho)^Tu_{0,l}\bigr)=K_{kl}$, exactly
Eq.~\eqref{eq:K-matrix-def}. Restricting to the $+\sigma_0$ branch
(assumed here nondegenerate with respect to $-\sigma_0$, which holds
throughout the example below) gives Eq.~\eqref{eq:svd-splitting}.
\end{proof}

\begin{corollary}[Nuclear-norm derivative]
\label{cor:nuclear-norm-first-order}
If $\sigma_0$ is the only singular value of $\mathcal M_S(\rho_0)$
(as is the case for $\GHZ_3$, \cite{aschauer2026a}), then
\begin{equation*}
\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}
\norm{\mathcal M_S(\rho(\varepsilon))}_*=\tr(K).
\end{equation*}
More generally this is the contribution of the $\sigma_0$-block alone;
non-degenerate singular values contribute their own
$u_{0}^T\mathcal M_S(\delta\rho)v_{0}$ term individually, and the total
derivative is the sum over all blocks.
\end{corollary}

\begin{remark}[Why a compression matrix, not a higher-order tensor]
\label{rem:nonsmoothness-lewis-sendov}
This answers directly the question that motivated this section: the
first-order object governing how a degenerate shadow singular value
splits is neither the shadow map itself (which stays linear, hence
featureless) nor a higher-order tensor, but a single real symmetric
$m\times m$ matrix per degenerate block --- an ordinary eigenvalue
problem, not an SVD. This is the shadow-map instance of the general
fact that unitarily invariant matrix norms such as $\norm{\cdot}_*$ are
differentiable at simple spectra but only \emph{directionally}
differentiable at repeated ones, with the one-sided directional
derivative controlled by exactly this kind of compressed, symmetrized
perturbation \cite{lewissendov2005}. Two mechanisms of this note make
degenerate blocks the rule rather than the exception:
Proposition~\ref{prop:block-diagonal} forces exact degeneracy whenever
$\rho_0$ has a compatible continuous symmetry, and
Lemma~\ref{lem:stabilizer-degeneracy} forces it combinatorially for
any Pauli-diagonal (stabilizer) state --- so every symmetric or
stabilizer state used numerically in this note sits at a kink of
$\norm{\mathcal M_S(\cdot)}_*$, and Theorem~\ref{thm:degenerate-svd-perturbation}
is the tool for resolving that kink in any chosen direction.
\end{remark}

\subsection{Exact rescaling under single-qubit Pauli dephasing}
\label{sec:pauli-dephasing-exact}

Before specializing Theorem~\ref{thm:degenerate-svd-perturbation} to an
example, it is worth isolating one perturbation direction for which
the first-order theory above turns out to be exact to \emph{all}
orders. The mechanism is a completely general, state-independent fact
about single-qubit Pauli dephasing, with no reference to symmetry or
to the stabilizer formalism.

\begin{proposition}[Exact diagonal rescaling under Pauli dephasing]
\label{prop:pauli-dephasing-exact}
Fix a qubit party $b\in P$ and a Pauli label $p\in\{1,2,3\}$
(i.e.\ $\sigma_p\in\{X,Y,Z\}$ on $b$). For $q\in[0,1]$ define the
two-outcome dephasing channel
\begin{equation*}
\Lambda_b^p(\rho;q):=q\,\rho+(1-q)\,\sigma_p^{(b)}\rho\,\sigma_p^{(b)}.
\end{equation*}
Then for every state $\rho$ and every Pauli string index
$\vec\imath=(i_1,\dots,i_n)\in\{0,1,2,3\}^n$,
\begin{equation}
\tr\bigl[\Lambda_b^p(\rho;q)\,\sigma_{\vec\imath}\bigr]
=w(i_b)\,\tr[\rho\,\sigma_{\vec\imath}],
\qquad
w(i_b):=\begin{cases}1,& i_b\in\{0,p\},\\ 2q-1,& i_b\in\{1,2,3\}\setminus\{p\},\end{cases}
\label{eq:pauli-dephasing-rescale}
\end{equation}
for every $q\in[0,1]$, exactly --- not merely to leading order in
$1-q$. Consequently every shadow map, sub-block, or sector profile of
$\Lambda_b^p(\rho;q)$ is obtained from the corresponding object for
$\rho$ by rescaling each Pauli-basis coefficient with $b$-leg index
$i_b$ anticommuting with $\sigma_p$ by the fixed factor $2q-1$, and
leaving every coefficient with $i_b$ commuting with $\sigma_p$
unchanged, for every $q\in[0,1]$.
\end{proposition}

\begin{proof}
Conjugation by the involution $\sigma_p^{(b)}$ satisfies
$\sigma_p^{(b)}\sigma_{\vec\imath}\sigma_p^{(b)}=\epsilon(i_b)\,\sigma_{\vec\imath}$,
with $\epsilon(i_b)=+1$ if the single-qubit factor $\sigma_{i_b}$
commutes with $\sigma_p$ (i.e.\ $i_b\in\{0,p\}$) and $\epsilon(i_b)=-1$
if it anticommutes ($i_b\in\{1,2,3\}\setminus\{p\}$). Hence
\begin{equation*}
\tr\bigl[\Lambda_b^p(\rho;q)\sigma_{\vec\imath}\bigr]
=q\,\tr[\rho\sigma_{\vec\imath}]+(1-q)\,\tr\bigl[\sigma_p^{(b)}\rho\,\sigma_p^{(b)}\sigma_{\vec\imath}\bigr]
=\bigl(q+(1-q)\epsilon(i_b)\bigr)\tr[\rho\sigma_{\vec\imath}],
\end{equation*}
using cyclicity of the trace. This is exactly
Eq.~\eqref{eq:pauli-dephasing-rescale}, since $q+(1-q)=1$ and
$q-(1-q)=2q-1$. Every shadow-map quantity in this note is a linear,
coordinate-wise function of these coefficients, so the claimed
rescaling of shadow maps follows immediately.
\end{proof}

\subsection{Worked example: dephasing the $\GHZ_3$ state}
\label{ex:ghz3-dephasing}

Recall from Corollary~\ref{cor:stabilizer-examples} above that for
$\rho_0=\lvert\GHZ_3\rangle\!\langle\GHZ_3\rvert$ and source party $A$,
the normalized map $\mathcal M_A(\rho_0)=\sigma_0U_0$ has the single
triply-degenerate singular value $\sigma_0=\sqrt{2/3}=\sqrt6/3$
(forced exactly by Lemma~\ref{lem:stabilizer-degeneracy} and
Corollary~\ref{cor:stabilizer-examples}, since $\GHZ_3$ carries no
continuous collective symmetry (cf. Remark~\ref{rem:no-universal-sharpening}),
with $V_0=\id_3$ (the standard $x,y,z$ source basis) and $U_0$ the
explicit orthonormal $15\times3$ matrix reproduced by
\texttt{scripts/ghz3\_shadow\_map\_symbolic.py}. We study the effect of
single-qubit dephasing on party $B$, i.e.\ the direction
$\delta\rho^{(P)}:=P_B\rho_0P_B-\rho_0$ for $P\in\{X,Y,Z\}$, physically
the dominant error channel ($Z$: $T_2$-type dephasing in the
computational/stabilizer basis) versus dephasing along an axis
transverse to it ($X$, $Y$).

\begin{lemma}[Diagonal perturbations rescale a degenerate block directly]
\label{lem:diagonal-perturbation}
In the setting of Theorem~\ref{thm:degenerate-svd-perturbation},
suppose $\mathcal M_S(\delta\rho)V_0=\sigma_0U_0D$ for some real
diagonal matrix $D=\diag(d_1,\dots,d_m)$. Then $K=\sigma_0D$ exactly,
so $\lambda_i(K)=\sigma_0d_i$.
\end{lemma}
\begin{proof}
$K=U_0^T\mathcal M_S(\delta\rho)V_0=U_0^T(\sigma_0U_0D)=\sigma_0(U_0^TU_0)D=\sigma_0D$,
using orthonormality of $U_0$; $D$ diagonal is already symmetric, so
no further symmetrization is needed.
\end{proof}

For $\delta\rho^{(Z)}$, every nonzero Pauli coefficient of $\rho_0$
with $A$-index $z$ (the two-body terms $ZZI,ZIZ$ of
Corollary~\ref{cor:stabilizer-examples}) has $B$-index in $\{0,z\}$,
commuting with $Z$; every nonzero coefficient with $A$-index $x$ or $y$
(the three-body terms $XXX,YXY,XYY,YYX$) has $B$-index in $\{x,y\}$,
anticommuting with $Z$. By Proposition~\ref{prop:pauli-dephasing-exact}
this is exactly the diagonal case of Lemma~\ref{lem:diagonal-perturbation}
with $D=\diag(-2,-2,0)$ (columns $x,y$ entirely anticommuting, column
$z$ entirely commuting), giving
\begin{equation}
K_Z=\sigma_0\diag(-2,-2,0)=\diag\Bigl(-\tfrac{2\sqrt6}3,\,-\tfrac{2\sqrt6}3,\,0\Bigr).
\label{eq:KZ-result}
\end{equation}
For $\delta\rho^{(X)}$ and $\delta\rho^{(Y)}$ the anticommuting and
commuting coefficients are no longer aligned with the $x,y$ vs.\ $z$
source columns (each column now mixes a commuting and an anticommuting
entry), so Lemma~\ref{lem:diagonal-perturbation} does not apply
directly; a direct evaluation of Eq.~\eqref{eq:K-matrix-def}
(reproduced exactly by \texttt{scripts/ghz3\_perturbation\_symbolic.py})
gives instead the isotropic results
\begin{equation}
K_X=K_Y=-\frac{\sqrt6}3\,\id_3.
\label{eq:KXY-result}
\end{equation}

\begin{table}[h]
\centering
\small
\begin{tabular}{@{}lll@{}}
\toprule
dephasing axis on $B$ & eigenvalues of $K$ & interpretation \\
\midrule
$Z$ (computational/stabilizer basis) & $-\tfrac{2\sqrt6}3,\,-\tfrac{2\sqrt6}3,\,0$ & $z$-channel exactly protected, $x,y$ decay twice as fast \\
$X$ & $-\tfrac{\sqrt6}3,\,-\tfrac{\sqrt6}3,\,-\tfrac{\sqrt6}3$ & isotropic decay, no protected direction \\
$Y$ & $-\tfrac{\sqrt6}3,\,-\tfrac{\sqrt6}3,\,-\tfrac{\sqrt6}3$ & isotropic decay, no protected direction \\
\bottomrule
\end{tabular}
\caption{First-order splitting of the triply-degenerate singular value
$\sigma_0=\sqrt6/3$ of $\mathcal M_A(\GHZ_3)$ under single-qubit
dephasing on party $B$, from Eqs.~\eqref{eq:KZ-result}--\eqref{eq:KXY-result}.
By Corollary~\ref{cor:nuclear-norm-first-order},
$\tr K_Z=-\tfrac{4\sqrt6}3$ and $\tr K_X=\tr K_Y=-\sqrt6$, matching a
direct finite-difference evaluation of $\norm{\mathcal M_A(\rho_0+
\varepsilon\delta\rho^{(P)})}_*$ to six digits at $\varepsilon=10^{-2},10^{-3}$.}
\label{tab:ghz3-dephasing}
\end{table}

The zero eigenvalue under $Z$-dephasing is not merely a first-order
artifact: it upgrades to an exact, non-perturbative statement once the
perturbation \emph{direction} $\delta\rho^{(Z)}$ is replaced by the
genuine one-parameter channel it generates.

\begin{corollary}[Exact protection of the $z$-channel, all orders]
\label{cor:ghz3-exact-channel}
Let $\rho(q):=\Lambda_B^{3}(\rho_0;q)=q\rho_0+(1-q)Z_B\rho_0Z_B$,
$q\in[0,1]$ (a valid state for every $q$, unlike the affine family
$\rho_0+\varepsilon\,\delta\rho^{(Z)}$, which leaves the state space
once $\varepsilon\notin[0,1]$). Then, exactly,
\begin{equation*}
\mathcal M_A(\rho(q))=\sigma_0\Bigl[(2q-1)\,u_xe_x^T+(2q-1)\,u_ye_y^T+1\cdot u_ze_z^T\Bigr],
\end{equation*}
already in singular value form, with singular values
\begin{equation*}
\sigma_x(q)=\sigma_y(q)=\lvert2q-1\rvert\,\sigma_0,
\qquad
\sigma_z(q)=\sigma_0,
\end{equation*}
for every $q\in[0,1]$. Writing $q=1-\varepsilon$ recovers
Eq.~\eqref{eq:svd-splitting} with $K_Z$ of Eq.~\eqref{eq:KZ-result} as
the exact leading term near $\varepsilon=0$, with no further
corrections until the singular values $\sigma_x,\sigma_y$ hit zero at
$q=1/2$.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:pauli-dephasing-exact}, every coefficient
with $A$-index $z$ is left unchanged and every coefficient with
$A$-index $x$ or $y$ is rescaled by $(2q-1)$, for every $q$ --- this is
exactly the source-column structure identified above, now exact rather
than first order. Since $U_0,V_0=\id_3$ already diagonalize
$\mathcal M_A(\rho_0)$ with equal singular value $\sigma_0$ on every
column, rescaling column $x,y$ by $(2q-1)$ and leaving column $z$ fixed
preserves this diagonal (singular-value) form exactly, with the stated
singular values.
\end{proof}

\begin{remark}[Why $Z$ is special, and the connection to noiseless subsystems]
The exactness in Corollary~\ref{cor:ghz3-exact-channel} is special to
$Z$-dephasing: it holds precisely because the stabilizer generators of
$\GHZ_3$ that carry the source index $z$ ($ZZI,ZIZ$) are themselves
built from $Z$ on every leg, hence commute with $Z_B$ termwise, while
every generator carrying source index $x$ or $y$ ($XXX,YXY,XYY,YYX$)
anticommutes with $Z_B$ uniformly. This is the same combinatorial
stabilizer mechanism as Lemma~\ref{lem:stabilizer-degeneracy}, now
controlling not just the degeneracy of $\mathcal M_A(\rho_0)$ but also
its exact response to a symmetry-breaking channel. Physically, the
$A\!\to\! B,C$ correlation carried by $Z_AZ_B$ and $Z_AZ_C$ is a
decoherence-free direction for exactly the noise that commutes with
it, in the sense familiar from the noiseless-subsystem and operator
quantum error correction literature
\cite{zanardirasetti1997,kribslaflammepoulin2005}: here the roles of
``logical'' information and ``noise operator'' are played by a
single fixed shadow-map correlation channel rather than by an encoded
qubit, but the underlying commutation argument is the same one. By
contrast, $X$- and $Y$-dephasing anticommute with two of the three
stabilizer generators carrying each source index and commute with the
third, so no single source column is uniformly protected or uniformly
decaying in the exact (all-$q$) sense; Eq.~\eqref{eq:KXY-result}
therefore describes only the leading-order behavior near $q=1$, not a
closed global formula of the type in Corollary~\ref{cor:ghz3-exact-channel}.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{A polynomial-size witness for permutation-symmetric cluster sources}
\label{sec:polynomial-witness}
%\label{sec:multinomial-collapse}

\cite{aschauer2026a} raised the cost of evaluating $\mathcal M_S(\rho)$
as a genuine obstruction once $|S|$ or $|S^c|$ grows. We now show that this
obstruction disappears entirely for a natural and large class of states: those
invariant under permuting the parties \emph{within} $S$ and \emph{within} $S^c$
separately. This is an instance of the weak, cut-factorizing hypothesis of
Proposition~\ref{prop:block-diagonal} --- not the strong, party-local one, since a
transposition of two parties inside $S$ is not of the form $U^{(a)}\otimes U^{(b)}$
(compare Remark~\ref{rem:permutation-separating-example}) --- specialized to the
full symmetric group, whose isotypic decomposition on a Pauli-string alphabet
collapses to an elementary orbit-counting argument.

\paragraph{Setup.} Fix qubits ($d_a=2$) for concreteness (Remark~\ref{rem:qudit-collapse}
below records the general-$d$ statement). Fix a source cluster $S$ with $|S|=m$
and let $S^c$ be its complement, $|S^c|=l:=n-m$. Consider the single bigraduated
block $M_{S\to S^c}(\rho)$ of Definition~\ref{def:bigraduated} with $V=S$, $T=S^c$
--- the ``fully active'' block already singled out after
Corollary~\ref{cor:sub-block} as the witness for genuine joint $S$-cluster
correlation. In the orthonormal Pauli-string basis this is the
$3^m\times 3^l$ matrix
\begin{equation*}
c(\vec\imath,\vec\jmath):=\tr\bigl(\rho\,\sigma_{\vec\imath}\otimes\sigma_{\vec\jmath}\bigr),
\qquad
\vec\imath\in\{x,y,z\}^m,\ \vec\jmath\in\{x,y,z\}^l.
\end{equation*}
For $\vec\imath\in\{x,y,z\}^m$ let $\mathrm{type}(\vec\imath)=(a_x,a_y,a_z)\in\mathcal T_m$,
$\mathcal T_m:=\{(a_x,a_y,a_z)\in\mathbb Z_{\ge0}^3: a_x+a_y+a_z=m\}$, record how
often each label occurs; the symmetric group $S_m$ acts on $\{x,y,z\}^m$ by
permuting positions, with orbits exactly the level sets of $\mathrm{type}$, and the
orbit of type $\alpha$ has size the multinomial coefficient $\binom m\alpha:=\frac{m!}{a_x!a_y!a_z!}$.
Note $|\mathcal T_m|=\binom{m+2}{2}$, polynomial in $m$, against the ambient $3^m$.

\begin{proposition}[Multinomial block collapse]
\label{prop:multinomial-collapse}
Suppose $\rho$ is invariant under $U_\pi\otimes U_{\pi'}$ for every $\pi\in S_m$
acting by the permutation unitary on $\mathcal H^{(S)}$ and every $\pi'\in S_l$
acting by the permutation unitary on $\mathcal H^{(S^c)}$. Then $c(\vec\imath,\vec\jmath)$
depends on $(\vec\imath,\vec\jmath)$ only through $(\mathrm{type}(\vec\imath),\mathrm{type}(\vec\jmath))\in\mathcal T_m\times\mathcal T_l$;
write $\tilde c(\alpha,\beta)$ for this common value, and define the
multinomial-weighted reduced matrix $\hat C\in\mathbb C^{|\mathcal T_m|\times|\mathcal T_l|}$ by
\begin{equation}
\hat C_{\alpha,\beta}:=\sqrt{\binom m\alpha\binom l\beta}\;\tilde c(\alpha,\beta).
\label{eq:reduced-dicke-matrix}
\end{equation}
Then $M_{S\to S^c}(\rho)$ and $\hat C$ have identical nonzero singular values
(the smaller matrix effectively zero-padded), so in particular
\begin{equation*}
\norm{M_{S\to S^c}(\rho)}_*=\norm{\hat C}_*,
\qquad
\norm{M_{S\to S^c}(\rho)}_{\fro}=\norm{\hat C}_{\fro}.
\end{equation*}
\end{proposition}

\begin{proof}
$U_\pi$-invariance of $\rho$ gives $\tr(\rho\,\sigma_{\pi\vec\imath}\otimes\sigma_{\vec\jmath})
=\tr(U_\pi^\dagger\rho U_\pi\,\sigma_{\vec\imath}\otimes\sigma_{\vec\jmath})=c(\vec\imath,\vec\jmath)$
for every $\pi\in S_m$, since conjugating $\rho$ by $U_\pi$ permutes the legs of
the Pauli string in the trace exactly as $\pi$ permutes $\vec\imath$; the analogous
identity on the target side follows from $U_{\pi'}$-invariance. Hence $c$ is
constant on $S_m$-orbit $\times$ $S_l$-orbit classes, proving the first claim.

For the matrix identity, let $u_\alpha\in\mathbb R^{3^m}$ be the normalized
indicator vector of the orbit of type $\alpha$ (equal to $1/\sqrt{\binom m\alpha}$
on that orbit and $0$ elsewhere), and $v_\beta\in\mathbb R^{3^l}$ analogously.
Since distinct orbits partition the index set, $U:=(u_\alpha)_{\alpha\in\mathcal T_m}$
and $V:=(v_\beta)_{\beta\in\mathcal T_l}$ have orthonormal columns. For
$\vec\imath$ of type $\alpha$ and $\vec\jmath$ of type $\beta$,
\begin{equation*}
(U\hat C V^T)_{\vec\imath,\vec\jmath}
=\frac{1}{\sqrt{\binom m\alpha}}\,\hat C_{\alpha,\beta}\,\frac1{\sqrt{\binom l\beta}}
=\tilde c(\alpha,\beta)=c(\vec\imath,\vec\jmath),
\end{equation*}
so $M_{S\to S^c}(\rho)=U\hat C\,V^T$. Left and right multiplication by
matrices with orthonormal columns preserves singular values up to zero-padding
(e.g.\ $M^TM=V\hat C^T\hat C V^T$ has the same nonzero eigenvalues as $\hat C^T\hat C$,
since $V^TV=\id$), giving the claim.
\end{proof}

\begin{corollary}[Normalized witness]
\label{cor:dicke-normalized}
If $\rho$ is separable across $S\mid S^c$, then, in the normalization of
Corollary~\ref{cor:sub-block} applied with $\mathcal V=\{S\}$, $\mathcal T=\{S^c\}$,
\begin{equation*}
\frac{\norm{\hat C}_*}{\sqrt{(2^m-1)(2^l-1)}}
=\norm{\widehat M_{S\to S^c}(\rho)}_*
\le\norm{\mathcal M_S(\rho)}_*\le1.
\end{equation*}
\end{corollary}

The point of Proposition~\ref{prop:multinomial-collapse} is computational, not a
sharper bound: it lets one evaluate $\norm{M_{S\to S^c}(\rho)}_*$ exactly via an
SVD of size $\binom{m+2}{2}\times\binom{l+2}{2}$ rather than $3^m\times3^l$, for
\emph{any} state with the stated permutation symmetry, with no approximation.

\begin{remark}[General local dimension]
\label{rem:qudit-collapse}
For general finite $d_a\equiv d$ on $S$ (and possibly different $d$ on $S^c$),
the same argument replaces $\{x,y,z\}$ by any fixed orthonormal generator set
of size $d^2-1$; orbits of $S_m$ on $(d^2-1)$-letter strings of length $m$ are
again indexed by compositions $\mathcal T_m^{(d)}=\{\alpha\in\mathbb Z_{\ge0}^{d^2-1}:\sum\alpha_i=m\}$,
of cardinality $\binom{m+d^2-2}{d^2-2}$ --- polynomial in $m$ for fixed $d$,
exponential in $d$ but that dependence is orthogonal to the point made here.
\end{remark}

\subsection{Worked example: Dicke states}
\label{sec:dicke-example}

\begin{lemma}[Closed-form Dicke correlator]
\label{lem:dicke-correlator}
Let $\lvert D_n^k\rangle:=\binom nk^{-1/2}\sum_{\lvert w\rvert=k}\lvert w\rangle$
be the $n$-qubit Dicke state of weight $k$, and let $P$ be a Pauli string of type
$(n_I,n_X,n_Y,n_Z)$ ($n_I+n_X+n_Y+n_Z=n$, no reference to which positions carry
which label needed, by Proposition~\ref{prop:multinomial-collapse}-type symmetry
applied to the full state). Then, if $n_X+n_Y$ is even,
\begin{equation}
\langle D_n^k\rvert P\lvert D_n^k\rangle
=\frac{i^{\,n_Y}}{\binom nk}
\sum_{\substack{a_X+a_Y=(n_X+n_Y)/2\\ a_I+a_Z=k-(n_X+n_Y)/2}}
\binom{n_X}{a_X}\binom{n_Y}{a_Y}\binom{n_I}{a_I}\binom{n_Z}{a_Z}(-1)^{a_Y+a_Z},
\label{eq:dicke-correlator}
\end{equation}
and $=0$ if $n_X+n_Y$ is odd.
\end{lemma}

\begin{proof}
Write $P\lvert w\rangle=\phi(w)\lvert w'\rangle$ where $w'$ flips the bits of $w$
at the $X$- and $Y$-labeled positions and leaves the rest unchanged, and
$\phi(w)=\prod_{Z\text{-pos}}(-1)^{w_i}\prod_{Y\text{-pos}}i(1-2w_i)$ (the $X$-
factors carry no phase). Since $\lvert D_n^k\rangle$ has support only on weight-$k$
strings, $\langle D_n^k\rvert P\lvert D_n^k\rangle=\binom nk^{-1}\sum_{\lvert w\rvert=k,\,\lvert w'\rvert=k}\phi(w)$.
The weight-preservation condition $\lvert w'\rvert=\lvert w\rvert$ forces exactly
half of the $n_X+n_Y$ flipped positions to hold a $1$ in $w$, i.e.\ $a_X+a_Y=(n_X+n_Y)/2$
if $w$ has $a_X$ ones among the $X$-positions and $a_Y$ ones among the
$Y$-positions (impossible if $n_X+n_Y$ is odd); the total-weight condition
$a_I+a_X+a_Y+a_Z=k$ then gives $a_I+a_Z=k-(n_X+n_Y)/2$. Summing $\phi(w)=(-1)^{a_Z}\cdot i^{\,n_Y-a_Y}(-i)^{a_Y}=(-1)^{a_Z+a_Y}i^{\,n_Y}$
over all $w$ with prescribed $(a_I,a_X,a_Y,a_Z)$, weighted by the number
$\binom{n_I}{a_I}\binom{n_X}{a_X}\binom{n_Y}{a_Y}\binom{n_Z}{a_Z}$ of such $w$, gives
Eq.~\eqref{eq:dicke-correlator}.
\end{proof}

Dicke states are invariant under the full party-permutation group $S_n$, hence
in particular under $S_m\times S_l$ for any cut $S\mid S^c$, so
Proposition~\ref{prop:multinomial-collapse} applies with $\tilde c(\alpha,\beta)$
given directly by Eq.~\eqref{eq:dicke-correlator} at $(n_I,n_X,n_Y,n_Z)=(n-m-l,\alpha_x+\beta_x,\alpha_y+\beta_y,\alpha_z+\beta_z)$.

\begin{example}
\label{ex:dicke-scaling}
For $n=9$, $k=4$, $S,S^c$ of size $m=4$, $l=3$, direct construction of the full
$81\times27$ matrix $M_{S\to S^c}(\rho)$ and of the reduced $15\times10$ matrix
$\hat C$ of Eq.~\eqref{eq:reduced-dicke-matrix} give identical singular-value
spectra to $10$ decimal digits, with $\norm{M_{S\to S^c}}_*=\norm{\hat C}_*=3.0972760203\ldots$,
confirming Proposition~\ref{prop:multinomial-collapse} numerically as well as
algebraically.

The practical payoff is in scaling $m,l$ far beyond what the ambient
$3^m\times3^l$ matrix could ever accommodate. Table~\ref{tab:dicke-scaling}
reports $\norm{\hat C}_*$ and wall-clock time for a fixed weight $k=3$ Dicke
state as $n,m,l$ grow; the reduced matrix and its exact SVD remain cheap
throughout, while the corresponding ambient matrix size is astronomically
out of reach already by $n\approx60$.

\begin{table}[h]
\centering
\small
\begin{tabular}{@{}rrrrrr@{}}
\toprule
$n$ & $m$ & $l$ & $\hat C$ size & ambient $3^m\times3^l$ & $\norm{\hat C}_*$ \\
\midrule
10  & 4  & 4  & $15\times15$   & $8.1\times10^{1}\times8.1\times10^{1}$  & $5.66$ \\
30  & 10 & 8  & $66\times45$   & $5.9\times10^{4}\times6.6\times10^{3}$  & $4.33$ \\
60  & 20 & 15 & $231\times136$ & $3.5\times10^{9}\times1.4\times10^{7}$  & $4.93$ \\
100 & 30 & 25 & $496\times351$ & $2.1\times10^{14}\times8.5\times10^{11}$ & $4.42$ \\
200 & 40 & 35 & $861\times666$ & $1.2\times10^{19}\times5.0\times10^{16}$ & $1.85$ \\
\bottomrule
\end{tabular}
\caption{Exact evaluation of $\norm{M_{S\to S^c}(\rho)}_*$ for the $k=3$ Dicke
state via the reduced matrix $\hat C$, Eq.~\eqref{eq:reduced-dicke-matrix}.
The rightmost ambient-matrix sizes are the dimensions an unreduced computation
would require; already at $n=200$ this is about $6\times10^{35}$ entries, versus
$861\times666\approx5.7\times10^{5}$ for $\hat C$ (computed, with full SVD, in
about $1.3$ seconds on a single core). The script
\texttt{scripts/dicke\_block\_collapse.py} reproduces this table and the exact
match with the ambient matrix at small $n$.}
\label{tab:dicke-scaling}
\end{table}

Two caveats: First, Table~\ref{tab:dicke-scaling}
reports the \emph{unnormalized} block $\norm{M_{S\to S^c}}_*$, not the
cut-separable witness $\norm{\widehat M_{S\to S^c}}_*$ of
Corollary~\ref{cor:dicke-normalized}; the latter divides by
$\sqrt{(2^m-1)(2^l-1)}$, which grows so quickly with $m,l$ that the normalized
witness is far below the separable threshold $1$ at the larger table entries ---
the large-$n$ rows demonstrate tractability of the computation, not entanglement
detection at that scale. Second, at half filling $k=n/2$ the unnormalized
correlator itself decays with $n$ (a genuine physical effect, not a numerical
artifact); Table~\ref{tab:dicke-scaling} therefore fixes a small $k=3$ so the
signal stays $O(1)$ throughout the range shown.
\end{example}




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\section{Outlook and open questions}
\label{sec:outlook}
%% offene Punkte (gemeinsamer Eigenbasis-Ansatz, systematischer Scan
% aller 4-Qubit-Graphen, % extremale biseparable Zustände, volle
% S_m-Symmetrie jenseits von TensorKit)

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\bibliographystyle{plain} 
\bibliography{references}

\end{document}