From 3aed2e91c77f557ab353ea539a2e308d6499c978 Mon Sep 17 00:00:00 2001 From: Hans Aschauer Date: Thu, 16 Jul 2026 21:52:29 +0200 Subject: [PATCH] revise abstract to introduce bigraduated shadow map and its implications for multipartite entanglement criteria --- paper/symmetric_shadow_maps_formal.tex | 102 ++++++++++++++++++++++++- 1 file changed, 100 insertions(+), 2 deletions(-) diff --git a/paper/symmetric_shadow_maps_formal.tex b/paper/symmetric_shadow_maps_formal.tex index 35e1107..6453aa9 100644 --- a/paper/symmetric_shadow_maps_formal.tex +++ b/paper/symmetric_shadow_maps_formal.tex @@ -35,7 +35,28 @@ \maketitle \begin{abstract} -The correlation strengths $L_S$ from the local-invariant correlation-sector approach of Aschauer, Calsamiglia, Hein, and Briegel \cite{aschauer} are simple and local-unitary invariant, but they compress each correlation block to a single Frobenius norm. We present a canonical source-indexed direct-sum response operator for multipartite systems of arbitrary finite local dimensions: fixing one source party $a$, we stack all correlation-response maps from $a$ into every nonempty correlation sector of its complement to obtain $\mathcal M_a(\rho)$, the combined shadow map. We prove that if $\rho$ is separable across the cut $a\mid\bar a$, then $\norm{\mathcal M_a(\rho)}_*\le 1$ after a natural dimension-dependent normalization. The construction extends verbatim from a single source party to an arbitrary source cluster $S$, with no change to the proof: the resulting bigraduated shadow map $\mathcal M_S(\rho)$ obeys the same bound across $S\mid S^c$, and every sub-block restriction of it is automatically a valid witness, which lets the four-qubit Smolin state's exact saturation at every $2\mid2$ cut sit alongside its clear $1\mid3$ violation within one framework. Averaging or maximizing the single-party maps over the source party yields source-aggregated cut-sensitive witness functionals, and in equal local dimensions these become permutation-invariant. We then work out the qubit case in detail. For three qubits the symmetric average admits an explicit biseparable threshold, giving a genuine tripartite entanglement criterion within this framework. Numerically, all $38$ connected labeled graph states on four qubits satisfy $\Phi_{\mathrm{sym}}=\Phi_{\max}=6/\sqrt 7$, crossing the fully separable white-noise threshold already at $p>\sqrt 7/6\approx 0.441$, while representative two-qubit marginals remain PPT in that region. Across the families $\GHZ_n$, line-graph, ring-graph, $W_n$, and D\"ur states for $n=3,4,5$, the graph-like families show a rigid common scaling pattern, the $W_n$ family lies slightly lower, and the D\"ur family falls below the fully separable benchmark for $n=4,5$. We then show that every construction in the note---single-party and cluster shadow maps alike, together with all their sub-block compressions---is a matricization or slice of one canonical full Bloch tensor, so that the recurring bound $\le 1$ is a single rank-one fact inherited unchanged through each linear operation rather than a family of separately proved statements. These results identify a structured mixed-order response architecture that is not visible in scalar sector sums or in a single unfolding of the full correlation tensor, and they suggest a natural bridge from explicit qubit criteria to a fuller correlation-moment hierarchy. +Multipartite correlation-tensor criteria for entanglement are well +established, including full-tensor unfoldings and cut-aware block +trace-norm bounds. We introduce a bigraduated shadow map +$\mathcal M_S(\rho)$ that stacks all correlation-response maps from a +source cluster $S$ into every sector of the complement, and show that +$\norm{\mathcal M_S(\rho)}_*\le1$ for states separable across $S\mid S^c$ +--- a bound that recovers, rather than supersedes, existing +correlation-matrix and realignment criteria. Its value lies in the +structure the bigraduation exposes: every sub-block and sector profile +is automatically a valid witness, though we show these profiles are not +invariant under unitaries on the coarse cut; the map itself is one +matricization of a single augmented Bloch tensor, so the bound is +inherited rather than separately proved at each level; and when $\rho$ +carries an exact or twirled symmetry, the map block-diagonalizes further +by representation-theoretic isotype, giving a second, combinatorial +(stabilizer-support) mechanism, together accounting for degeneracies +observed numerically throughout. We give an explicit three-qubit +biseparable threshold, detect the two-qutrit Tiles bound entangled state +despite it being numerically PPT, and show all $38$ connected +four-qubit graph states saturate a common value $6/\sqrt7$, robust to +white noise past $p\approx0.441$, undetected by any two-qubit marginal. +The four-qubit biseparable threshold remains open. \end{abstract} \section{From the old criterion to a new object} @@ -81,8 +102,71 @@ M_{a\to T}, \end{equation*} into one canonical operator $\mathcal M_a(\rho)$. +The following table makes this division explicit, separating what is already available in the cited literature from what this note adds on top of it. + +\begin{table}[h] +\centering +\small +\begin{tabular}{@{}p{0.46\textwidth}p{0.46\textwidth}@{}} +\toprule +\textbf{Already established elsewhere} & \textbf{Reference} \\ +\midrule +A single collapsed correlation-matrix / realignment-type nuclear-norm bound for one bipartite cut & \cite{devicente,chenwu,devicentehuber,hassanjoag} \\ +Cut-aware, mixed-order block trace-norm criteria built from an extended correlation tensor & \cite{liyaoyangfei2025} \\ +Averaging a bipartition-indexed trace norm over many bipartitions to obtain a genuine multipartite criterion & \cite{liyaoyangfei2025}, Theorem~5 \\ +\bottomrule +\end{tabular} +\vspace{0.8em} +\begin{tabular}{@{}p{0.46\textwidth}p{0.46\textwidth}@{}} +\toprule +\textbf{New in this note} & \textbf{Where} \\ +\midrule +Simultaneous bigraduation of the same cut matrix by \emph{both} source sector $V$ and target sector $T$, rather than one collapsed block & Definition~\ref{def:bigraduated} \\ +Sub-block and sector-profile witnesses that this bigraduation makes available automatically, with no separate proof & Corollary~\ref{cor:sub-block}, profiles $\Phi_{a\to T}$, $\Phi^{(2)}_{a\to T}$ \\ +Sector profiles are demonstrably \emph{not} invariant under unitaries acting collectively on the coarse complement, even though the full cut norm is & Proposition following Corollary~\ref{cor:projection-bound}, Bell-product/CNOT example \\ +Every shadow map, at every graduation level, is one matricization of a single augmented Bloch tensor, so the $\le1$ bound is inherited rather than reproved at each level & Theorem~\ref{thm:one-fact}, Section~\ref{sec:tensor-viewpoint} \\ +An explicit, fully proved three-qubit biseparable threshold in closed form & Eq.~\eqref{eq:bisep-threshold} \\ +Two rigorously distinguished degeneracy mechanisms (continuous isotypic symmetry vs.\ discrete stabilizer support) explaining numerically observed singular-value degeneracies & Section~\ref{sec:symmetry-blocks} \\ +A qutrit PPT-entangled benchmark and a systematic scan of $38$ four-qubit graph states and $n=3,4,5$ state families & qutrit benchmark in Section~\ref{sec:tensor-viewpoint}, Section~\ref{sec:qubit-numerics} \\ +\bottomrule +\end{tabular} +\caption{What this note recovers from the existing correlation-tensor literature (top) versus what it contributes on top of that baseline (bottom). The headline bound $\norm{\mathcal M_S(\rho)}_*\le1$ itself belongs to the top half; the substantive claims of the note are the structural and numerical items in the bottom half.} +\label{tab:novelty} +\end{table} + A guiding thread throughout the note is that this architecture is not tied to a single source party. Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sources} show that fixing a source \emph{cluster} $S\subseteq P$ instead of a single party costs nothing in the proof: the rank-one mechanism behind the cut-separable bound is agnostic to how many parties sit on the source side. Section~\ref{sec:tensor-viewpoint} then makes precise in what sense this is not a coincidence: every construction in this note---the single-party map $\mathcal M_a$, its cluster generalization $\mathcal M_S$, and every sub-block compression of either---is a matricization or sub-block restriction of one and the same full correlation tensor, and the bound $\le 1$ is a single rank-one fact about that tensor, inherited unchanged through each linear operation. The single-party case is simply the version of this fact that is cheapest to state first. +As a concrete demonstration that this stacked structure detects entanglement invisible to the standard PPT test, Section~\ref{sec:tensor-viewpoint} exhibits a two-qutrit state built from the Tiles unextendible product basis of \cite{bennettUPB}: it is numerically PPT to machine precision, so the Peres--Horodecki criterion is silent on it \cite{peres,horodeckiPPT}, yet the shadow-map witness detects its entanglement outright. We flag this example here because it is, in our view, the single clearest piece of evidence in the note that the construction has practical bite beyond reorganizing known bounds. + +\subsection*{Notation guide} + +The construction accumulates several closely related maps and scalars as it is refined step by step; Table~\ref{tab:notation} collects the main ones for reference, in the order they are introduced. + +\begin{table}[h] +\centering +\small +\begin{tabular}{@{}lp{0.72\textwidth}@{}} +\toprule +Symbol & Meaning \\ +\midrule +$C_S(\rho)$, $L_S(\rho)$ & sector correlation tensor and its squared Frobenius norm (Aschauer \emph{et al.} \cite{aschauer}) \\ +$\widetilde{\mathcal M}_a(\rho)$ & intrinsic (basis-free) one-vs-rest response map, source party $a$ \\ +$M_{a\to T}(\rho)$ & unnormalized response block landing in target sector $T\subseteq\bar a$ \\ +$\mathcal M_a(\rho)$ & combined, normalized shadow map for a single source party $a$ (Definition~\ref{def:combined-shadow}) \\ +$\widehat M_{a\to T}(\rho)$ & sector shadow map, normalized independently of the other sectors \\ +$\Phi_{a\to T}(\rho)$, $\Phi^{(2)}_{a\to T}(\rho)$ & sector nuclear and Frobenius shadow profiles, $\norm{\widehat M_{a\to T}}_*$ and $\norm{\widehat M_{a\to T}}_\fro^2$ \\ +$\Phi_a^{(\le k)}(\rho)$, $\Phi_a^{(\ge k)}(\rho)$ & nuclear norm after projecting onto grouped target sectors of order $\le k$ or $\ge k$ \\ +$\mathcal M_S(\rho)$ & bigraduated shadow map for a source \emph{cluster} $S$ (Definition~\ref{def:bigraduated}); reduces to $\mathcal M_a(\rho)$ when $S=\{a\}$ \\ +$M_{V\to T}(\rho)$ & bigraduated block from source sector $V\subseteq S$ to target sector $T\subseteq S^c$ \\ +$\Phi_{\mathrm{sym}}(\rho)$, $\Phi_{\max}(\rho)$ & source-aggregated functionals, average and maximum of $\norm{\mathcal M_a(\rho)}_*$ over $a\in P$ \\ +$\mathcal C(\rho)$ & full order-$n$ Bloch tensor of which every map above is a matricization or slice (Definition~\ref{def:full-tensor}) \\ +$A_\lambda$ & reduced shadow map on the multiplicity space of isotype $\lambda$, once $\rho$ carries a compatible symmetry (Section~\ref{sec:symmetry-blocks}) \\ +\bottomrule +\end{tabular} +\caption{Main notation, in order of introduction. $\Phi_{\mathrm{sym}}$, $\Phi_{\max}$, and the biseparable threshold are qubit-specific; everything above them in the table is defined for arbitrary finite local dimensions.} +\label{tab:notation} +\end{table} + \section{One-vs-rest response maps (shadow maps)} \label{sec:response-maps} @@ -568,6 +652,11 @@ i.e.\ the matrix representation of $\widetilde{\mathcal M}_S(\rho)$, normalized For $S=\{a\}$, Definition~\ref{def:bigraduated} reduces exactly to Definition~\ref{def:combined-shadow}: the source-side decomposition then has only the single summand $V=S$, so the bigraduation collapses to the ordinary target-only graduation of $\mathcal M_a$. Definition~\ref{def:combined-shadow} is thus the singleton case of this construction, not a separate object introduced in parallel to it. +\begin{remark}[Computational cost] +\label{rem:computational-cost} +The total number of scalar entries in $\mathcal M_S(\rho)$ is $(d_S-1)(d_{S^c}-1)$, exactly as for the collapsed cut matrix, so evaluating the norm $\norm{\mathcal M_S(\rho)}_*$ in Theorem~\ref{thm:cluster-cut} costs a single singular value decomposition of that size and is no more expensive than the realignment-type bounds it recovers. The bigraduation of Definition~\ref{def:bigraduated} does not change this cost; it only reorganizes the same entries into $(2^{|S|}-1)(2^{|S^c|}-1)$ combinatorial blocks $M_{V\to T}$. Consequently, using the full map $\mathcal M_S(\rho)$ as a single witness remains cheap, but exploiting the sub-block witnesses of Corollary~\ref{cor:sub-block} exhaustively --- inspecting every source sector against every target sector separately, rather than only the handful used in the examples below --- requires examining up to $O(2^{|S|+|S^c|})=O(2^n)$ individual blocks in the worst case. Section~\ref{sec:symmetry-blocks} shows that when $\rho$ carries a compatible symmetry, this exponential proliferation of combinatorial blocks collapses instead onto a typically much smaller number of representation-theoretic multiplicity spaces, giving a tractable route to the same sub-block information without enumerating every sector by hand. +\end{remark} + \begin{theorem}[Cluster cut-separable bound] \label{thm:cluster-cut} Let $\rho$ be separable across the cut $S\mid S^c$. Then @@ -1036,6 +1125,15 @@ Let $G$ be a compact group acting on the system by local unitaries, $g\mapsto \b \end{equation} where $\lambda$ ranges over the irreducible representations of $G$ appearing on either side, $V_\lambda$ denotes a fixed model of the irreducible representation of dimension $d_\lambda$, and $M_\lambda^{(S)}$, $M_\lambda^{(S^c)}$ are the corresponding multiplicity spaces. This is exactly the same type of decomposition already used for the source and target sector gradings in Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sources}, now taken with respect to a representation-theoretic rather than a combinatorial grading; the two coincide only in special cases. +These two gradings are not independent structures competing for the same +role: Proposition~\ref{prop:sector-symmetry-compatible} below shows that +each correlation sector $\V_V^{(S)}$ is itself $G$-invariant, so that the +isotypic decomposition refines the sector decomposition rather than +sitting alongside it. We record the block-diagonality statement first, +since it is the one with the direct entanglement-detection consequence, +and return to the compatibility of the two gradings --- and the resulting +joint refinement of the bigraduated shadow map --- immediately afterward. + 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] @@ -1273,4 +1371,4 @@ There is also a natural exact continuation of this program. In the qubit case, t \bibliographystyle{plain} \bibliography{references} -\end{document} +\end{document} \ No newline at end of file