From 1552fad8418185528a501d39516eec4c428a3459 Mon Sep 17 00:00:00 2001 From: Hans Aschauer Date: Sun, 2 Aug 2026 17:29:52 +0200 Subject: [PATCH] fix: refine language and clarify symmetry-related concepts in the shadow map paper --- paper/shadow_maps_symmetric_states.tex | 90 ++++++++++++++++++++++---- 1 file changed, 79 insertions(+), 11 deletions(-) diff --git a/paper/shadow_maps_symmetric_states.tex b/paper/shadow_maps_symmetric_states.tex index ca7e2a6..3aa2364 100644 --- a/paper/shadow_maps_symmetric_states.tex +++ b/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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%