fix: refine language and clarify symmetry-related concepts in the shadow map paper

paper/shadow_maps_symmetric_states.tex

@@ -182,8 +182,8 @@ 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
+combinatorial blocks in the worst case, as noted in \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 $3^m\times3^l$
@@ -527,7 +527,7 @@ numerically identical to the cut bound in \cite{aschauer2026a}, but now a statem
 \subsection{Axial $U(1)$ symmetry: an exact worked example}
 \label{sec:u1-example}
 
-The isotypic mechanism of Proposition~\ref{prop:block-diagonal} is easiest to see concretely for an abelian symmetry group, where it reduces to an ordinary charge-conservation selection rule.
+The isotypic mechanism of Proposition~\ref{prop:block-diagonal} is easiest to see concretely for an abelian symmetry group, where it reduces to an ordinary charge-conservation selection rule. That a global $U(1)$ (or other abelian) symmetry splits the singular values of a bipartite correlation-type matrix into charge sectors is by now a recurring theme in the entanglement-detection literature under the name \emph{symmetry-resolved entanglement}: it has been used to sharpen partial-transpose-moment witnesses \cite{nevenetal2021}, to decompose the singular values of the realignment matrix itself into $U(1)$-charge blocks \cite{rathetal2023}, and, in a very recent and structurally close construction, to build symmetry-resolved bounds on the negativity directly from charge-sector-resolved realignment-matrix singular values \cite{tarabungahaug2025}. The mechanism below is the same charge-conservation selection rule applied instead to the bigraduated shadow map $\mathcal M_S(\rho)$, worked out completely and in closed form for one illustrative four-qubit family rather than as a general witness-improvement scheme.
 
 \begin{remark}[Abelian symmetries as selection rules]
 \label{rem:abelian-selection-rule}
@@ -960,11 +960,25 @@ closed global formula of the type in Corollary~\ref{cor:ghz3-exact-channel}.
 \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
+As noted in \cite{aschauer2026a}, the cost of evaluating $\mathcal M_S(\rho)$
+is a genuine obstruction once $|S|$ or $|S^c|$ grows. That a compatible
+permutation symmetry removes exactly this kind of exponential obstruction is
+itself well established in the entanglement-detection literature: permutation-
+invariant quantum tomography reconstructs an $n$-qubit state from $O(n^3)$
+rather than $O(4^n)$ settings \cite{tothetal2010pitomography,moroderhyllustoth2012},
+and, closer to genuine multipartite entanglement specifically, the PPT-mixture
+SDP used for GME detection collapses from exponential to polynomial size for
+permutationally invariant states \cite{novomorodergühne2013}, with dedicated
+closed-form criteria available for Dicke states in particular
+\cite{bergmanngühne2013}. All of these results work in the coupled-spin
+(Clebsch--Gordan) basis adapted to the \emph{full} $S_n$-invariance of the
+state itself. We show that the same qualitative phenomenon --- exponential
+collapsing to polynomial --- also holds directly for the bigraduated shadow
+map of a single fixed cut $S\mid S^c$, under only the weaker hypothesis that
+$\rho$ is invariant under $S_m\times S_l$ separately on each side of the cut,
+via an elementary orbit-counting argument on Pauli strings that needs no
+representation theory of $SU(2)$ beyond the symmetric-group orbit structure
+itself. 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
@@ -1165,9 +1179,63 @@ signal stays $O(1)$ throughout the range shown.
 
 \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)
+
+Three threads from this note remain open, and connect back to the
+research program of the companion note~\cite{aschauer2026a} as
+recorded there.
+
+First, a \emph{joint eigenbasis} combining both degeneracy mechanisms
+of Section~\ref{sec:mechanisms} has been used here only block by block
+(Corollary~\ref{cor:joint-refinement}), never assembled into a single
+explicit basis simultaneously adapted to the combinatorial sector
+grading and the isotypic decomposition for a state that carries both
+structures at once. Constructing this joint eigenbasis explicitly, and
+determining whether it exposes further forced degeneracies beyond
+those already accounted for by the two mechanisms separately, is left
+open.
+
+Second, a \emph{systematic scan over all connected four-qubit graphs}
+--- not just the $38$ states already reported in \cite{aschauer2026a}
+--- using the stabilizer mechanism of Lemma~\ref{lem:stabilizer-degeneracy}
+together with the automorphism-driven weak-hypothesis refinements of
+Example~\ref{ex:ring-reflection}, could in principle classify every
+forced degeneracy pattern across the whole family from the two
+mechanisms of this note alone, without a separate numerical SVD per
+graph. Carrying this out completely, and comparing the result against
+the existing numerical scan, is left for future work.
+
+Third, on the computational side, Section~\ref{sec:polynomial-witness}
+exploits only the $U(1)$-type or full-$S_m$ permutation symmetry of a
+single cluster's own parties. Using the same symmetry constraints
+instead to \emph{search} for extremal biseparable states --- states
+that saturate a cut-separable or GME bound of \cite{aschauer2026a}
+subject to a prescribed collective symmetry --- would turn the
+degeneracy mechanisms of Sections~\ref{sec:mechanisms} into a
+constructive search tool rather than only an explanatory or
+verification one. A full treatment of this direction, together with
+the extension of Proposition~\ref{prop:multinomial-collapse} beyond
+the $U(1)$ block structure used in Section~\ref{sec:u1-example} to the
+full $S_m$ permutation symmetry of a source cluster on the target
+side simultaneously, is left to future work; the qudit and multi-cluster
+generalizations noted in Remark~\ref{rem:qudit-collapse} would need to
+be revisited in that setting as well.
+
+Finally, a broader positioning remark. This note and its companion
+\cite{aschauer2026a} study the singular-value structure of one
+particular unfolding of the Bloch correlation tensor, the bigraduated
+shadow map $\mathcal M_S(\rho)$, chosen for its direct cut-separable
+threshold at nuclear norm $1$. A structurally related but
+distinct program studies the \emph{geometry} of the singular values of
+a bipartite correlation matrix itself --- e.g.\ which singular-value
+profiles are achievable at a given Schmidt number, accessed via
+randomized measurements rather than full state tomography
+\cite{wyderkaketterer2023}. The two programs share an underlying
+object (Bloch-correlation-matrix singular values) but ask different
+questions of it --- a threshold for one fixed functional versus the
+achievable range of the full spectrum --- and reconciling the two
+viewpoints, e.g.\ asking what the shadow-map degeneracy mechanisms of
+Section~\ref{sec:mechanisms} imply for the achievable singular-value
+geometry in that sense, is a natural further question not pursued here.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%