feat: add new scripts for combined S_m and SO(3) symmetry checks and general-r correlation tensor validation

paper/shadow_maps_symmetric_states.tex

@@ -731,17 +731,20 @@ independent of which cut's matricization is subsequently taken.
 $T(\rho)(\vec\imath)=\tr(\rho\,\sigma_{\vec\imath})$ is linear in $\rho$; substituting $\rho=\sum_{a,b}c_{ab}\ketbra{\phi_a}{\phi_b}$ and using $\tr(\ketbra{\phi_a}{\phi_b}\sigma_{\vec\imath})=\bra{\phi_b}\sigma_{\vec\imath}\ket{\phi_a}=T_{ba}(\vec\imath)$ gives Eq.~\eqref{eq:general-coherence-decomposition} termwise; the cut dependence enters only afterward, through the reshape of the index set $\vec\imath$, and does not affect the coefficients $c_{ab}$ or the tensors $T_{ab}$.
 \end{proof}
 
-% TODO (open / not yet carried out): Proposition~\ref{prop:coherence-templates}
-% is stated and used above only for r=2 real generators (psi_1, psi_2), where
-% the single real cross term C_{12} suffices because the c_{ab} happen to be
-% real and symmetric in that example. For general r this requires r(r+1)/2
-% independent real tensors (r diagonal T_aa plus the real/imaginary parts of
-% the r(r-1)/2 off-diagonal overlaps, or equivalently just track the full
-% Hermitian r x r coefficient matrix c_{ab} against the T_{ab} disregarding
-% the a<->b symmetry) -- this is mechanical but has not been written out or
-% numerically tested here for r>2. Do this before submission if the general
-% statement is kept; otherwise restate the Proposition for r=2 only and move
-% the general case to a remark.
+\begin{corollary}[Real-tensor count, corrected]
+\label{cor:real-tensor-count}
+Since $\sigma_{\vec\imath}$ is Hermitian, $T_{ba}=\overline{T_{ab}}$; in particular each $T_{aa}$ is real. Writing $T_{ab}=P_{ab}+iQ_{ab}$ for $a<b$, Eq.~\eqref{eq:general-coherence-decomposition} becomes, for Hermitian $c$,
+\begin{equation}
+T(\rho) = \sum_{a=1}^r c_{aa}\,T_{aa} \;+\; \sum_{a<b}\Bigl(2\,\mathrm{Re}(c_{ab})\,P_{ab} \;-\; 2\,\mathrm{Im}(c_{ab})\,Q_{ab}\Bigr),
+\label{eq:real-tensor-count}
+\end{equation}
+a real-linear combination of exactly $r+2\binom r2=r^2$ fixed real tensors $\{T_{aa}\}\cup\{P_{ab},Q_{ab}\}_{a<b}$ -- matching the real dimension of the space of Hermitian $r\times r$ matrices, as it must, since $\rho\mapsto T(\rho)$ is a real-linear injection (the $T_{ab}$ are linearly independent whenever the $\ket{\phi_a}$ are) from that space. (For $r=2$ this is exactly Eq.~\eqref{eq:coherence-decomposition}, with $C_{12}=T_{12}+T_{21}=2P_{12}$ and $Q_{12}=0$ there because $c_{12}=\cos\alpha\sin\alpha$ was taken real in that example.)
+\end{corollary}
+
+\begin{example}[$r=3$, genuinely mixed]
+\label{ex:general-r-three}
+Take $\ket{\phi_1},\ket{\phi_2}$ as in Eq.~\eqref{eq:singlet-network-states} together with a third perfect matching $\ket{\phi_3}\propto\ket{\psi^-}_{AF}\ket{\psi^-}_{BD}\ket{\psi^-}_{CE}$; all three are collectively invariant, with pairwise overlaps $\braket{\phi_a}{\phi_b}=1/4$ ($a\ne b$) and Gram matrix of condition number $2$ (linearly independent). For a Haar-random Hermitian PSD $c$ of full rank $3$ (not proportional to a rank-$1$ projector, i.e.\ $\rho=\Phi c\Phi^\dagger/\tr(\Phi c\Phi^\dagger)$ a genuinely mixed state with three distinct nonzero eigenvalues), Eq.~\eqref{eq:general-coherence-decomposition} -- built once from the $9=3^2$ tensors $T_{ab}$ -- reproduces the brute-force correlation tensor of $\rho$ to $10^{-16}$, at both the $ABC\mid DEF$ and $AB\mid CDEF$ cuts simultaneously, with no further contraction over the $64$-dimensional Hilbert space (\texttt{scripts/general\_r\_check.py}). This extends the verification of Section~\ref{sec:cut-independent-data} from a pure two-state superposition to a genuinely mixed three-state case, as Proposition~\ref{prop:coherence-templates} and Corollary~\ref{cor:real-tensor-count} require.
+\end{example}
 
 \begin{remark}[What this does and does not establish]
 \label{rem:recoupling-scope}
@@ -811,15 +814,49 @@ verified exactly (sympy, symbolic Clebsch--Gordan coefficients) to satisfy $(-1)
 Eq.~\eqref{eq:xi-sum} is a special value of a Racah recoupling coefficient with one of the six angular momenta equal to $1$, a case with classically known closed forms (see e.g.\ Varshalovich, \emph{Quantum Theory of Angular Momentum}, tables of $6j$ symbols with a unit argument); we verify the needed closed form directly by exhaustive exact symbolic evaluation for $p,j\le3$ rather than by matching a specific textbook formula, since two attempts at identifying the exact literature convention (transcribed as comments in \texttt{scripts/search\_6j.py}) did not reproduce it and the discrepancy was not tracked down. The proof above is therefore complete and independently checked (symbolically for the $\Xi$-sum, and end-to-end against brute-force simulation of both example states to $10^{-15}$), but does not give a citation-ready closed form for general $(p,j)$ beyond $p,j\le3$; extending the exact symbolic check to arbitrary $p,j$, or locating the precise textbook identity, is left for later. The conjugation phase of Eq.~\eqref{eq:conjugation-phase} is likewise verified rather than derived from first principles for general $n_\ell$; Step 2 sketches the expected inductive argument.
 \end{remark}
 
-% TODO (outlook, not attempted): combining the S_m-permutation collapse of
-% Section~\ref{sec:polynomial-witness} (Proposition~\ref{prop:multinomial-collapse})
-% with the SO(3) branching of this subsection. For a cluster invariant under
-% BOTH S_m and collective SO(3) simultaneously, Sym^m(R^3) is known classically
-% to decompose multiplicity-free into spherical harmonics, m_j=1 for
-% j=m,m-2,...; combined with Proposition~\ref{prop:multinomial-collapse} this
-% should collapse a full sector to O(m) one-dimensional channels rather than
-% the polynomial-but-not-linear \binom{m+2}{2} of that Proposition alone.
-% Not worked out or tested here.
+\subsubsection{Combining with permutation symmetry: multiplicity-free channels}
+\label{sec:sm-so3-combination}
+
+% ============================================================
+% NEW SUBSUBSECTION -- draft. Resolves the second open point from an
+% earlier draft's outlook TODO: combining the S_m-permutation collapse of
+% Proposition~\ref{prop:multinomial-collapse} with the SO(3) branching of
+% this subsection. Verified computationally on a concrete 6-qubit example
+% (two coupled 3-qubit Dicke/symmetric multiplets) in
+% scripts/combined_sm_so3_collapse.py.
+% ============================================================
+
+The two reduction mechanisms developed in this note -- the $S_m$-permutation collapse of Proposition~\ref{prop:multinomial-collapse} and the $SO(3)$ branching of this subsection -- combine multiplicatively rather than merely coexisting, and the combination is worth making explicit because it is genuinely stronger than either alone.
+
+\begin{proposition}[Harmonic decomposition of symmetric tensors; classical]
+\label{prop:harmonic-decomposition}
+As an $SO(3)$-representation, $\mathrm{Sym}^m(\mathbb R^3)$ decomposes multiplicity-free,
+\begin{equation}
+\mathrm{Sym}^m(\mathbb R^3) \;\cong\; \bigoplus_{k=0}^{\lfloor m/2\rfloor} \mathcal H_{m-2k},
+\qquad \dim\mathcal H_j = 2j+1,
+\label{eq:harmonic-decomposition}
+\end{equation}
+where $\mathcal H_j$ denotes the space of degree-$j$ solid harmonics (equivalently, the traceless part of $\mathrm{Sym}^j(\mathbb R^3)$). This is the standard trace-decomposition of symmetric tensors underlying the multipole expansion; see e.g.\ Fulton--Harris.
+\end{proposition}
+
+Consequently, for a source cluster $S$ of size $m$ that is invariant under \emph{both} $S_m$ (permutations of its own $m$ parties) \emph{and} the collective hypothesis~\eqref{eq:collective-hypothesis}, the source sector $\mathcal V_S^{(S)}$ collapses in two independent, compatible steps rather than one: first from $3^m$ to $\binom{m+2}{2}$ via Proposition~\ref{prop:multinomial-collapse} (a polynomial, but quadratic, reduction), and then, \emph{within} that $\binom{m+2}2$-dimensional space, the isotypic multiplicities collapse to exactly $1$ for each of the $O(m)$ surviving values $j=m,m-2,\dots$ -- dramatically finer than the generic branching multiplicities $m_j^{(m)}$ of Proposition~\ref{prop:branching-su2}, which grow with $m$ (e.g.\ $m_1^{(3)}=3$ unrestricted, versus multiplicity exactly $1$ within $\mathrm{Sym}^3$). Computationally, this second step costs one further Hermitian eigendecomposition of the $J^2$ Casimir restricted to the already-small $\binom{m+2}2$-dimensional space -- i.e.\ diagonalizing a matrix of side length $\binom{m+2}2$, not $3^m$ -- after which every surviving channel carries a genuine \emph{scalar} reduced matrix element $A_j$ (no residual multiplicity, hence no Clebsch--Gordan bookkeeping of the kind needed in Section~\ref{sec:six-j-recoupling}).
+
+\begin{example}[A doubly-symmetric six-qubit state]
+\label{ex:dicke-network}
+Let $S=\{A,B,C\}$, $S^c=\{D,E,F\}$, and let $\ket{D_3^k}$ denote the $3$-qubit Dicke state of weight $k$ (Section~\ref{sec:dicke-example}). The state
+\begin{equation}
+\ket{Q} \;\propto\; \sum_{k=0}^3 (-1)^k\, \ket{D_3^k}_{ABC}\otimes\ket{D_3^{3-k}}_{DEF}
+\label{eq:dicke-network-state}
+\end{equation}
+is the canonical invariant ("singlet") combination of the two spin-$3/2$ multiplets spanned by the Dicke states on $ABC$ and on $DEF$ respectively (directly analogous to the two-spin-$1$ singlet construction of Example~\ref{ex:aligned-singlets}, now for the \emph{physical} qubit spin instead of the Bloch-vector generator spin). By construction $\ket Q$ is $S_3$-symmetric separately on $ABC$ and on $DEF$ (each factor is built from Dicke states), and, being the canonical invariant combination of two matching total-spin multiplets, satisfies the full collective hypothesis~\eqref{eq:collective-hypothesis} (verified directly: $\lvert\bra{Q}U_g^{\otimes6}\ket{Q}\rvert=1$ to machine precision for random $g\in SU(2)$).
+
+Restricting the $27\times27$ block $M_{S\to S^c}(\ketbra QQ)$ to the $10$-dimensional symmetric subspace $\mathrm{Sym}^3(\mathbb C^3)\subset\mathcal V_S^{(S)}$ on both sides (via the orthonormal "type" basis $\{u_\alpha\}_{\alpha\in\mathcal T_3}$ of Proposition~\ref{prop:multinomial-collapse}) and diagonalizing the restricted Casimir $U^\dagger J^2_{\mathrm{tot}}U$ gives eigenvalues $2$ (three-fold, $j=1$) and $12$ (seven-fold, $j=3$) \emph{exactly}, with $j=0,2$ entirely absent -- confirming Proposition~\ref{prop:harmonic-decomposition} concretely for $m=3$ ($\dim\mathcal H_3+\dim\mathcal H_1=7+3=10$). The two surviving channels carry the scalar reduced matrix elements
+\begin{equation}
+A_1 = \tfrac13, \qquad A_3 = 2,
+\label{eq:dicke-network-values}
+\end{equation}
+satisfying $3A_1+7A_3=1+14=15=\norm{M_{S\to S^c}(\ketbra QQ)}_*$ exactly, matching the nuclear norm of the full, unrestricted $27\times27$ block computed directly -- so for this state \emph{every} unit of correlation captured by the cut already lives inside the doubly-symmetric sector (verified: the nuclear norm of $M_{S\to S^c}$ orthogonally projected \emph{away} from $\mathrm{Sym}^3\otimes\mathrm{Sym}^3$ is zero to $10^{-14}$).
+\end{example}
 
 \subsection{Exact degeneracy from stabilizer structure}