add script for multinomial block collapse and Dicke state scaling verification

paper/symmetric_shadow_maps_formal.tex

@@ -1215,7 +1215,8 @@ i.e.\ that the $2\times2$ array of blocks is circulant --- a prediction that fol
 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
+\end{equation*}
+\begin{equation*}
 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.
@@ -1412,6 +1413,208 @@ The $y$-parity grading occasionally useful for real-in-the-computational-basis s
 Proposition~\ref{prop:block-diagonal} and Lemma~\ref{lem:stabilizer-degeneracy} are complementary, not competing, and apply under disjoint hypotheses. The representation-theoretic mechanism applies whenever $\rho$ is genuinely $G$-invariant under some compact group $G$ acting by local unitaries and preserving the cut, regardless of whether $\rho$ is Pauli-diagonal; it says nothing about states, such as a generic finite-group-symmetric state built from a permutation representation, that are not Pauli-diagonal. The stabilizer mechanism applies whenever $\rho$ is (a uniform mixture over) a stabilizer code state, regardless of whether it possesses any continuous symmetry at all --- as is the case for $\GHZ_n$, the Smolin state, and every graph state used elsewhere in this note, none of which is invariant under a nontrivial continuous collective symmetry. In the (comparatively narrow) overlap where a state is both $G$-invariant for some continuous $G$ and Pauli-diagonal, both mechanisms apply and constrain the same block structure from different directions; outside that overlap, exactly one of the two is available, and it is this Lemma, not Proposition~\ref{prop:block-diagonal}, that accounts for every numerically observed degeneracy reported so far in this note.
 \end{remark}
 
+\subsection*{A polynomial-size witness for permutation-symmetric cluster sources}
+\label{sec:multinomial-collapse}
+
+Remark~\ref{rem:computational-cost} 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
+full symmetric group, whose isotypic decomposition on a Pauli-string alphabet
+collapses to an elementary orbit-counting argument.
+
+\paragraph{Setup.} Fix qubits ($d_a=2$) for concreteness (Remark~\ref{rem:qudit-collapse}
+below records the general-$d$ statement). Fix a source cluster $S$ with $|S|=m$
+and let $S^c$ be its complement, $|S^c|=l:=n-m$. Consider the single bigraduated
+block $M_{S\to S^c}(\rho)$ of Definition~\ref{def:bigraduated} with $V=S$, $T=S^c$
+--- the ``fully active'' block already singled out after
+Corollary~\ref{cor:sub-block} as the witness for genuine joint $S$-cluster
+correlation. In the orthonormal Pauli-string basis this is the
+$3^m\times 3^l$ matrix
+\begin{equation*}
+c(\vec\imath,\vec\jmath):=\tr\bigl(\rho\,\sigma_{\vec\imath}\otimes\sigma_{\vec\jmath}\bigr),
+\qquad
+\vec\imath\in\{x,y,z\}^m,\ \vec\jmath\in\{x,y,z\}^l.
+\end{equation*}
+For $\vec\imath\in\{x,y,z\}^m$ let $\mathrm{type}(\vec\imath)=(a_x,a_y,a_z)\in\mathcal T_m$,
+$\mathcal T_m:=\{(a_x,a_y,a_z)\in\mathbb Z_{\ge0}^3: a_x+a_y+a_z=m\}$, record how
+often each label occurs; the symmetric group $S_m$ acts on $\{x,y,z\}^m$ by
+permuting positions, with orbits exactly the level sets of $\mathrm{type}$, and the
+orbit of type $\alpha$ has size the multinomial coefficient $\binom m\alpha:=\frac{m!}{a_x!a_y!a_z!}$.
+Note $|\mathcal T_m|=\binom{m+2}{2}$, polynomial in $m$, against the ambient $3^m$.
+
+\begin{proposition}[Multinomial block collapse]
+\label{prop:multinomial-collapse}
+Suppose $\rho$ is invariant under $U_\pi\otimes U_{\pi'}$ for every $\pi\in S_m$
+acting by the permutation unitary on $\mathcal H^{(S)}$ and every $\pi'\in S_l$
+acting by the permutation unitary on $\mathcal H^{(S^c)}$. Then $c(\vec\imath,\vec\jmath)$
+depends on $(\vec\imath,\vec\jmath)$ only through $(\mathrm{type}(\vec\imath),\mathrm{type}(\vec\jmath))\in\mathcal T_m\times\mathcal T_l$;
+write $\tilde c(\alpha,\beta)$ for this common value, and define the
+multinomial-weighted reduced matrix $\hat C\in\mathbb C^{|\mathcal T_m|\times|\mathcal T_l|}$ by
+\begin{equation}
+\hat C_{\alpha,\beta}:=\sqrt{\binom m\alpha\binom l\beta}\;\tilde c(\alpha,\beta).
+\label{eq:reduced-dicke-matrix}
+\end{equation}
+Then $M_{S\to S^c}(\rho)$ and $\hat C$ have identical nonzero singular values
+(the smaller matrix effectively zero-padded), so in particular
+\begin{equation*}
+\norm{M_{S\to S^c}(\rho)}_*=\norm{\hat C}_*,
+\qquad
+\norm{M_{S\to S^c}(\rho)}_{\fro}=\norm{\hat C}_{\fro}.
+\end{equation*}
+\end{proposition}
+
+\begin{proof}
+$U_\pi$-invariance of $\rho$ gives $\tr(\rho\,\sigma_{\pi\vec\imath}\otimes\sigma_{\vec\jmath})
+=\tr(U_\pi^\dagger\rho U_\pi\,\sigma_{\vec\imath}\otimes\sigma_{\vec\jmath})=c(\vec\imath,\vec\jmath)$
+for every $\pi\in S_m$, since conjugating $\rho$ by $U_\pi$ permutes the legs of
+the Pauli string in the trace exactly as $\pi$ permutes $\vec\imath$; the analogous
+identity on the target side follows from $U_{\pi'}$-invariance. Hence $c$ is
+constant on $S_m$-orbit $\times$ $S_l$-orbit classes, proving the first claim.
+
+For the matrix identity, let $u_\alpha\in\mathbb R^{3^m}$ be the normalized
+indicator vector of the orbit of type $\alpha$ (equal to $1/\sqrt{\binom m\alpha}$
+on that orbit and $0$ elsewhere), and $v_\beta\in\mathbb R^{3^l}$ analogously.
+Since distinct orbits partition the index set, $U:=(u_\alpha)_{\alpha\in\mathcal T_m}$
+and $V:=(v_\beta)_{\beta\in\mathcal T_l}$ have orthonormal columns. For
+$\vec\imath$ of type $\alpha$ and $\vec\jmath$ of type $\beta$,
+\begin{equation*}
+(U\hat C V^T)_{\vec\imath,\vec\jmath}
+=\frac{1}{\sqrt{\binom m\alpha}}\,\hat C_{\alpha,\beta}\,\frac1{\sqrt{\binom l\beta}}
+=\tilde c(\alpha,\beta)=c(\vec\imath,\vec\jmath),
+\end{equation*}
+so $M_{S\to S^c}(\rho)=U\hat C\,V^T$. Left and right multiplication by
+matrices with orthonormal columns preserves singular values up to zero-padding
+(e.g.\ $M^TM=V\hat C^T\hat C V^T$ has the same nonzero eigenvalues as $\hat C^T\hat C$,
+since $V^TV=\id$), giving the claim.
+\end{proof}
+
+\begin{corollary}[Normalized witness]
+\label{cor:dicke-normalized}
+If $\rho$ is separable across $S\mid S^c$, then, in the normalization of
+Corollary~\ref{cor:sub-block} applied with $\mathcal V=\{S\}$, $\mathcal T=\{S^c\}$,
+\begin{equation*}
+\frac{\norm{\hat C}_*}{\sqrt{(2^m-1)(2^l-1)}}
+=\norm{\widehat M_{S\to S^c}(\rho)}_*
+\le\norm{\mathcal M_S(\rho)}_*\le1.
+\end{equation*}
+\end{corollary}
+
+The point of Proposition~\ref{prop:multinomial-collapse} is computational, not a
+sharper bound: it lets one evaluate $\norm{M_{S\to S^c}(\rho)}_*$ exactly via an
+SVD of size $\binom{m+2}{2}\times\binom{l+2}{2}$ rather than $3^m\times3^l$, for
+\emph{any} state with the stated permutation symmetry, with no approximation.
+
+\begin{remark}[General local dimension]
+\label{rem:qudit-collapse}
+For general finite $d_a\equiv d$ on $S$ (and possibly different $d$ on $S^c$),
+the same argument replaces $\{x,y,z\}$ by any fixed orthonormal generator set
+of size $d^2-1$; orbits of $S_m$ on $(d^2-1)$-letter strings of length $m$ are
+again indexed by compositions $\mathcal T_m^{(d)}=\{\alpha\in\mathbb Z_{\ge0}^{d^2-1}:\sum\alpha_i=m\}$,
+of cardinality $\binom{m+d^2-2}{d^2-2}$ --- polynomial in $m$ for fixed $d$,
+exponential in $d$ but that dependence is orthogonal to the point made here.
+\end{remark}
+
+\subsection*{Worked example: Dicke states}
+\label{sec:dicke-example}
+
+\begin{lemma}[Closed-form Dicke correlator]
+\label{lem:dicke-correlator}
+Let $\lvert D_n^k\rangle:=\binom nk^{-1/2}\sum_{\lvert w\rvert=k}\lvert w\rangle$
+be the $n$-qubit Dicke state of weight $k$, and let $P$ be a Pauli string of type
+$(n_I,n_X,n_Y,n_Z)$ ($n_I+n_X+n_Y+n_Z=n$, no reference to which positions carry
+which label needed, by Proposition~\ref{prop:multinomial-collapse}-type symmetry
+applied to the full state). Then, if $n_X+n_Y$ is even,
+\begin{equation}
+\langle D_n^k\rvert P\lvert D_n^k\rangle
+=\frac{i^{\,n_Y}}{\binom nk}
+\sum_{\substack{a_X+a_Y=(n_X+n_Y)/2\\ a_I+a_Z=k-(n_X+n_Y)/2}}
+\binom{n_X}{a_X}\binom{n_Y}{a_Y}\binom{n_I}{a_I}\binom{n_Z}{a_Z}(-1)^{a_Y+a_Z},
+\label{eq:dicke-correlator}
+\end{equation}
+and $=0$ if $n_X+n_Y$ is odd.
+\end{lemma}
+
+\begin{proof}
+Write $P\lvert w\rangle=\phi(w)\lvert w'\rangle$ where $w'$ flips the bits of $w$
+at the $X$- and $Y$-labeled positions and leaves the rest unchanged, and
+$\phi(w)=\prod_{Z\text{-pos}}(-1)^{w_i}\prod_{Y\text{-pos}}i(1-2w_i)$ (the $X$-
+factors carry no phase). Since $\lvert D_n^k\rangle$ has support only on weight-$k$
+strings, $\langle D_n^k\rvert P\lvert D_n^k\rangle=\binom nk^{-1}\sum_{\lvert w\rvert=k,\,\lvert w'\rvert=k}\phi(w)$.
+The weight-preservation condition $\lvert w'\rvert=\lvert w\rvert$ forces exactly
+half of the $n_X+n_Y$ flipped positions to hold a $1$ in $w$, i.e.\ $a_X+a_Y=(n_X+n_Y)/2$
+if $w$ has $a_X$ ones among the $X$-positions and $a_Y$ ones among the
+$Y$-positions (impossible if $n_X+n_Y$ is odd); the total-weight condition
+$a_I+a_X+a_Y+a_Z=k$ then gives $a_I+a_Z=k-(n_X+n_Y)/2$. Summing $\phi(w)=(-1)^{a_Z}\cdot i^{\,n_Y-a_Y}(-i)^{a_Y}=(-1)^{a_Z+a_Y}i^{\,n_Y}$
+over all $w$ with prescribed $(a_I,a_X,a_Y,a_Z)$, weighted by the number
+$\binom{n_I}{a_I}\binom{n_X}{a_X}\binom{n_Y}{a_Y}\binom{n_Z}{a_Z}$ of such $w$, gives
+Eq.~\eqref{eq:dicke-correlator}.
+\end{proof}
+
+Dicke states are invariant under the full party-permutation group $S_n$, hence
+in particular under $S_m\times S_l$ for any cut $S\mid S^c$, so
+Proposition~\ref{prop:multinomial-collapse} applies with $\tilde c(\alpha,\beta)$
+given directly by Eq.~\eqref{eq:dicke-correlator} at $(n_I,n_X,n_Y,n_Z)=(n-m-l,\alpha_x+\beta_x,\alpha_y+\beta_y,\alpha_z+\beta_z)$.
+
+\begin{example}
+\label{ex:dicke-scaling}
+For $n=9$, $k=4$, $S,S^c$ of size $m=4$, $l=3$, direct construction of the full
+$81\times27$ matrix $M_{S\to S^c}(\rho)$ and of the reduced $15\times10$ matrix
+$\hat C$ of Eq.~\eqref{eq:reduced-dicke-matrix} give identical singular-value
+spectra to $10$ decimal digits, with $\norm{M_{S\to S^c}}_*=\norm{\hat C}_*=3.0972760203\ldots$,
+confirming Proposition~\ref{prop:multinomial-collapse} numerically as well as
+algebraically.
+
+The practical payoff is in scaling $m,l$ far beyond what the ambient
+$3^m\times3^l$ matrix could ever accommodate. Table~\ref{tab:dicke-scaling}
+reports $\norm{\hat C}_*$ and wall-clock time for a fixed weight $k=3$ Dicke
+state as $n,m,l$ grow; the reduced matrix and its exact SVD remain cheap
+throughout, while the corresponding ambient matrix size is astronomically
+out of reach already by $n\approx60$.
+
+\begin{table}[h]
+\centering
+\small
+\begin{tabular}{@{}rrrrrr@{}}
+\toprule
+$n$ & $m$ & $l$ & $\hat C$ size & ambient $3^m\times3^l$ & $\norm{\hat C}_*$ \\
+\midrule
+10  & 4  & 4  & $15\times15$   & $8.1\times10^{1}\times8.1\times10^{1}$  & $5.66$ \\
+30  & 10 & 8  & $66\times45$   & $5.9\times10^{4}\times6.6\times10^{3}$  & $4.33$ \\
+60  & 20 & 15 & $231\times136$ & $3.5\times10^{9}\times1.4\times10^{7}$  & $4.93$ \\
+100 & 30 & 25 & $496\times351$ & $2.1\times10^{14}\times8.5\times10^{11}$ & $4.42$ \\
+200 & 40 & 35 & $861\times666$ & $1.2\times10^{19}\times5.0\times10^{16}$ & $1.85$ \\
+\bottomrule
+\end{tabular}
+\caption{Exact evaluation of $\norm{M_{S\to S^c}(\rho)}_*$ for the $k=3$ Dicke
+state via the reduced matrix $\hat C$, Eq.~\eqref{eq:reduced-dicke-matrix}.
+The rightmost ambient-matrix sizes are the dimensions an unreduced computation
+would require; already at $n=200$ this is about $6\times10^{35}$ entries, versus
+$861\times666\approx5.7\times10^{5}$ for $\hat C$ (computed, with full SVD, in
+about $1.3$ seconds on a single core). The script
+\texttt{scripts/dicke\_block\_collapse.py} reproduces this table and the exact
+match with the ambient matrix at small $n$.}
+\label{tab:dicke-scaling}
+\end{table}
+
+Two caveats keep this example honest. First, Table~\ref{tab:dicke-scaling}
+reports the \emph{unnormalized} block $\norm{M_{S\to S^c}}_*$, not the
+cut-separable witness $\norm{\widehat M_{S\to S^c}}_*$ of
+Corollary~\ref{cor:dicke-normalized}; the latter divides by
+$\sqrt{(2^m-1)(2^l-1)}$, which grows so quickly with $m,l$ that the normalized
+witness is far below the separable threshold $1$ at the larger table entries ---
+the large-$n$ rows demonstrate tractability of the computation, not entanglement
+detection at that scale. Second, at half filling $k=n/2$ the unnormalized
+correlator itself decays with $n$ (a genuine physical effect, not a numerical
+artifact); Table~\ref{tab:dicke-scaling} therefore fixes a small $k=3$ so the
+signal stays $O(1)$ throughout the range shown.
+\end{example}
+
+
 \section{Outlook}
 
 The combined shadow map should be viewed as a structured refinement of the older correlation strengths $L_S$ introduced by Aschauer \emph{et al.} \cite{aschauer}. Those quantities keep one Frobenius norm per tensor block; the present construction keeps the common one-vs-rest channel structure across all orthogonal sectors on the complement. In that sense it preserves the geometric spirit of that local-invariant sector decomposition while extracting more information from the same correlation data.