add example section for ring graph state and its symmetry properties

paper/symmetric_shadow_maps_formal.tex

@@ -17,6 +17,7 @@
 \newtheorem{corollary}{Corollary}
 \newtheorem{remark}{Remark}
 \newtheorem{lemma}{Lemma}
+\newtheorem{example}{Example}
 
 \newcommand{\tr}{\operatorname{tr}}
 \newcommand{\id}{\mathbbm{1}}
@@ -1114,9 +1115,12 @@ The constructions of Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sourc
 
 What this refinement does \emph{not} deliver is a numerically sharper detection threshold: Remark~\ref{rem:no-universal-sharpening} below shows that no universal, state-independent improvement over Theorems~\ref{thm:cut-bound} and \ref{thm:cluster-cut} exists at the level of an individual representation-theoretic block, except in the single-isotype case where the improvement is one of concentration rather than of threshold value.
 
-\subsection*{Setup}
 
-Let $G$ be a compact group acting on the system by local unitaries, $g\mapsto \bigotimes_{a\in P}U_g^{(a)}$, and suppose $\rho$ is $G$-invariant: $(\bigotimes_a U_g^{(a)})\,\rho\,(\bigotimes_a U_g^{(a)})^\dagger=\rho$ for all $g\in G$. Fix a cut $S\mid S^c$ preserved by $G$ as a set partition, so that $G$ acts on $\V_0^{(S)}$ and on $\V_0^{(S^c)}$ separately, via the adjoint representations $\mathrm{Ad}^{(S)}_g$, $\mathrm{Ad}^{(S^c)}_g$. Decompose both traceless spaces into isotypic components,
+\subsection*{Setup: two tiers of hypothesis}
+
+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
@@ -1125,20 +1129,15 @@ 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.
+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 hypotheses 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,
+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,
@@ -1159,7 +1158,7 @@ Proposition~\ref{prop:block-diagonal} organizes $\widetilde{\mathcal M}_S(\rho)$
 
 \begin{proposition}[Compatibility of sector and symmetry decompositions]
 \label{prop:sector-symmetry-compatible}
-Let $G$ act on the system by local unitaries, $g\mapsto\bigotimes_{a\in P}U_g^{(a)}$, 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
+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.
@@ -1193,6 +1192,60 @@ for reduced maps $A_{V,T,\lambda}$ on the corresponding multiplicity spaces. In
 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 Section~\ref{sec:qubit-numerics} 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 Section~\ref{sec:qubit-numerics}), 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}),
+\qquad
+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 Section~\ref{sec:qubit-numerics} 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]