paper/references.bib

@@ -370,77 +370,6 @@
   doi = {10.1103/PhysRevA.88.012305}
 }
 
-@article{tothetal2010pitomography,
-  author = {T{\'o}th, G{\'e}za and Wieczorek, Witlef and Gross, David and Krischek, Roland and Schwemmer, Christian and Weinfurter, Harald},
-  title = {Permutationally Invariant Quantum Tomography},
-  journal = {Physical Review Letters},
-  volume = {105},
-  number = {25},
-  pages = {250403},
-  year = {2010},
-  doi = {10.1103/PhysRevLett.105.250403}
-}
-
-@article{moroderhyllustoth2012,
-  author = {Moroder, Tobias and Hyllus, Philipp and T{\'o}th, G{\'e}za and Schwemmer, Christian and Niggebaum, Alexander and Gaile, Sarah and G{\"u}hne, Otfried and Weinfurter, Harald},
-  title = {Permutationally invariant state reconstruction},
-  journal = {New Journal of Physics},
-  volume = {14},
-  pages = {105001},
-  year = {2012},
-  doi = {10.1088/1367-2630/14/10/105001}
-}
-
-@article{bergmanngühne2013,
-  author = {Bergmann, Marcus and G{\"u}hne, Otfried},
-  title = {Entanglement criteria for Dicke states},
-  journal = {Journal of Physics A: Mathematical and Theoretical},
-  volume = {46},
-  pages = {385304},
-  year = {2013},
-  doi = {10.1088/1751-8113/46/38/385304}
-}
-
-@article{nevenetal2021,
-  author = {Neven, Antoine and Carrasco, Jose and Vitale, Vittorio and Kokail, Christian and Elben, Andreas and Dalmonte, Marcello and Calabrese, Pasquale and Zoller, Peter and Vermersch, Benoit and Kueng, Richard and Kraus, Barbara},
-  title = {Symmetry-resolved entanglement detection using partial transpose moments},
-  journal = {npj Quantum Information},
-  volume = {7},
-  pages = {152},
-  year = {2021},
-  doi = {10.1038/s41534-021-00487-y}
-}
-
-@article{rathetal2023,
-  author = {Rath, Aniket and Vitale, Vittorio and Murciano, Sara and Votto, Matteo and Dubail, Jerome and Kueng, Richard and Branciard, Cyril and Calabrese, Pasquale and Vermersch, Benoit},
-  title = {Entanglement Barrier and its Symmetry Resolution: Theory and Experiment},
-  journal = {PRX Quantum},
-  volume = {4},
-  pages = {010318},
-  year = {2023},
-  doi = {10.1103/PRXQuantum.4.010318}
-}
-
-@misc{tarabungahaug2025,
-  author = {Tarabunga, Poetri Sonya and Haug, Tobias},
-  title = {Quantifying mixed-state entanglement via partial transpose and realignment moments},
-  year = {2025},
-  eprint = {2507.13840},
-  archivePrefix = {arXiv},
-  primaryClass = {quant-ph},
-  url = {https://arxiv.org/abs/2507.13840}
-}
-
-@article{wyderkaketterer2023,
-  author = {Wyderka, Nikolai and Ketterer, Andreas},
-  title = {Probing the Geometry of Correlation Matrices with Randomized Measurements},
-  journal = {PRX Quantum},
-  volume = {4},
-  pages = {020325},
-  year = {2023},
-  doi = {10.1103/PRXQuantum.4.020325}
-}
-
 @article{jungnitschmorodergühne2011graphstates,
   author = {Jungnitsch, Bastian and Moroder, Tobias and G{\"u}hne, Otfried},
   title = {Entanglement Witnesses for Graph States: General Theory and Examples},

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, as noted in \cite{aschauer2026a}.
-Section~\ref{sec:polynomial-witness} shows this
+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 $3^m\times3^l$
@@ -527,7 +527,7 @@ numerically identical to the cut bound in \cite{aschauer2026a}, but now a statem
 \subsection{Axial $U(1)$ symmetry: an exact worked example}
 \label{sec:u1-example}
 
-The isotypic mechanism of Proposition~\ref{prop:block-diagonal} is easiest to see concretely for an abelian symmetry group, where it reduces to an ordinary charge-conservation selection rule. That a global $U(1)$ (or other abelian) symmetry splits the singular values of a bipartite correlation-type matrix into charge sectors is by now a recurring theme in the entanglement-detection literature under the name \emph{symmetry-resolved entanglement}: it has been used to sharpen partial-transpose-moment witnesses \cite{nevenetal2021}, to decompose the singular values of the realignment matrix itself into $U(1)$-charge blocks \cite{rathetal2023}, and, in a very recent and structurally close construction, to build symmetry-resolved bounds on the negativity directly from charge-sector-resolved realignment-matrix singular values \cite{tarabungahaug2025}. The mechanism below is the same charge-conservation selection rule applied instead to the bigraduated shadow map $\mathcal M_S(\rho)$, worked out completely and in closed form for one illustrative four-qubit family rather than as a general witness-improvement scheme.
+The isotypic mechanism of Proposition~\ref{prop:block-diagonal} is easiest to see concretely for an abelian symmetry group, where it reduces to an ordinary charge-conservation selection rule.
 
 \begin{remark}[Abelian symmetries as selection rules]
 \label{rem:abelian-selection-rule}
@@ -960,25 +960,11 @@ closed global formula of the type in Corollary~\ref{cor:ghz3-exact-channel}.
 \label{sec:polynomial-witness}
 %\label{sec:multinomial-collapse}
 
-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
+\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
@@ -1179,63 +1165,9 @@ signal stays $O(1)$ throughout the range shown.
 
 \section{Outlook and open questions}
 \label{sec:outlook}
-
-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.
+%% 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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