From 991af9956266e5faee8f27c93b10ba4b51c049cf Mon Sep 17 00:00:00 2001 From: Hans Aschauer Date: Tue, 11 Aug 2026 14:12:47 +0200 Subject: [PATCH] doc: include review comments --- paper/shadow_maps_symmetric_states.tex | 219 ++++++++++++++++++------- 1 file changed, 156 insertions(+), 63 deletions(-) diff --git a/paper/shadow_maps_symmetric_states.tex b/paper/shadow_maps_symmetric_states.tex index d932426..f9905e5 100644 --- a/paper/shadow_maps_symmetric_states.tex +++ b/paper/shadow_maps_symmetric_states.tex @@ -50,13 +50,20 @@ much finer block structure than a single collapsed correlation matrix. Several features of that structure --- the triple degeneracy of $\mathcal M_a(\GHZ_3)$'s singular values, the analogous triple degeneracy of the Smolin state under both the $1\mid3$ and $2\mid2$ cut, -and the common saturating value $6/\sqrt7$ shared by all $38$ connected +and the common saturating value $6/\sqrt7$ of the single-party ($1\mid3$) +cut functionals $\Phi_{\mathrm{sym}}=\Phi_{\max}$ shared by all $38$ connected four-qubit graph states --- were reported there as separate numerical facts. We show these degeneracies are forced \emph{exactly}, by two structurally independent mechanisms. The first is representation-theoretic: whenever $\rho$ is invariant under a compact group acting by unitaries compatible with the cut, Schur's lemma forces $\mathcal M_S(\rho)$ to be -block diagonal over isotypic components; we give the precise (weak, +block diagonal over isotypic components -- with each diagonal block itself +a scalar multiple of the identity on the real-type irreducible +representations that occur throughout the $SO(3)$ and permutation +mechanisms of this note (automatic there; the abelian $U(1)$ mechanism +below is of complex, not real, type and is handled instead by +complexifying and working with $1$-dimensional complex weight spaces +directly, sidestepping rather than using this reduction); we give the precise (weak, cut-factorizing versus strong, party-local) hypotheses under which this is compatible with the existing sector grading, and exhibit an axial-$U(1)$ four-qubit example, not itself a stabilizer state, where this mechanism @@ -79,7 +86,22 @@ permutation group of its constituent parties: the corresponding shadow-map block reduces, without approximation, from a $3^m\times3^l$ matrix to one of polynomial size $\binom{m+2}2\times\binom{l+2}2$, which we demonstrate for Dicke states with cluster sizes up to $m=40$, $l=35$ -within an $n=200$-qubit state. +within an $n=200$-qubit state. We then work out the representation +theory of the full, non-abelian collective hypothesis in detail: an +exact branching formula for the $SU(2)$ multiplicities, exact isotypic +projectors via the Casimir operator (no group averaging), an exact +recoupling formula relating the reduced blocks of two different +cuts of the same invariant tensor to one another by an explicit +dimension-ratio factor -- proved in general except for one finite +identity verified exhaustively and exactly for the six-qubit case +treated here rather than in closed form for arbitrary cluster size --, +a cut-independent template decomposition that +lets every cut and every state in a fixed invariant subspace be +recovered from $r^2$ fixed tensors without revisiting the full Hilbert +space, and, combining this with the permutation collapse, a further +multiplicity-free reduction to genuinely scalar reduced matrix elements +on $O(m)$ channels -- all illustrated on explicit six-qubit examples and +backed by exact symbolic and numerical verification throughout. \end{abstract} @@ -194,6 +216,28 @@ clusters of up to $m=40$, $l=35$ qubits within an $n=200$-qubit state (Table~\ref{tab:dicke-scaling}), far beyond what the ambient matrix could represent explicitly. +\emph{What if the collective symmetry is the full, non-abelian $SU(2)$, +not just its abelian $U(1)$ subgroup?} Section~\ref{sec:full-su2-symmetry} +develops this case in detail, going beyond the closed-form worked example +of Section~\ref{sec:u1-example}: an exact branching formula for the +resulting multiplicities (Proposition~\ref{prop:branching-su2}), exact +isotypic projectors via the $J^2$ Casimir rather than group averaging +(Corollary~\ref{cor:projector-norm-formula}), an exact recoupling +formula relating the reduced blocks of two different cuts of the same +invariant tensor by an explicit, closed-form factor -- proved in general +except for one finite identity that is verified exhaustively and exactly +for the six-qubit case treated here, rather than derived in closed form +for arbitrary cluster size +(Theorem~\ref{thm:six-j-recoupling}, Remark~\ref{rem:six-j-scope}), a cut-independent template +decomposition letting every cut and every state in a fixed invariant +subspace be recovered from finitely many fixed tensors without +revisiting the full Hilbert space (Proposition~\ref{prop:coherence-templates}), +and, combining this with the permutation collapse of +Section~\ref{sec:polynomial-witness}, a further reduction to genuinely +multiplicity-free, scalar reduced matrix elements +(Section~\ref{sec:sm-so3-combination}) -- all illustrated on explicit +six-qubit examples. + Section~\ref{sec:setup} fixes notation and recalls, without reproof, the minimal apparatus from~\cite{aschauer2026a} needed below: the sector decomposition of $\V_0^{(S)}$, $\V_0^{(S^c)}$, the definition of @@ -298,7 +342,7 @@ With $d_S:=\prod_{a\in S}d_a$ and $d_{S^c}:=\prod_{a\in S^c}d_a$, the \label{eq:bigraduated-map} \end{equation} i.e.\ the matrix of $\widetilde{\mathcal M}_S(\rho)$ in sector-adapted -orthonormal coordinates on both sides. For a singleton source $S=\{a\}$, +orthonormal coordinates on both sides. (Since the generators satisfy $\tr(\sigma_i\sigma_j)=d\,\delta_{ij}$, not $\delta_{ij}$, the raw $\sigma_i$ are Hilbert--Schmidt \emph{orthogonal} but not unit-normalized, so "orthonormal coordinates" means the basis $\{\sigma_i/\sqrt d\}$; the prefactor in Eq.~\eqref{eq:bigraduated-map} is exactly what converts matrix entries computed in the raw, unnormalized Pauli-string basis -- the basis in which every explicit computation and every worked example in this note, including the stabilizer mechanism of Section~\ref{sec:mechanisms} and all numerical scripts, is actually carried out -- into this orthonormal normalization.) For a singleton source $S=\{a\}$, the source-side decomposition~\eqref{eq:source-bloch-decomposition} has only the summand $V=S$, and \eqref{eq:bigraduated-map} reduces to the ordinary, target-only combined shadow map $\mathcal M_a(\rho)$ used @@ -335,7 +379,9 @@ projections. If $\rho$ is separable across $S\mid S^c$, then \norm{P_{\mathcal T}\,\mathcal M_S(\rho)\,P_{\mathcal V}}_*\ \le\ \norm{\mathcal M_S(\rho)}_*\ \le\ 1, \end{equation*} with no separate proof required: orthogonal projections are contractive -for the operator norm. +for the operator norm ($\norm{P}_{\mathrm{op}}\le1$), and the nuclear norm +is submultiplicative with respect to such contractions on each side, +$\norm{AXB}_*\le\norm{A}_{\mathrm{op}}\norm{X}_*\norm{B}_{\mathrm{op}}$, giving the stated bound with $A=P_{\mathcal T}$, $X=\mathcal M_S(\rho)$, $B=P_{\mathcal V}$. \end{corollary} Everything in Sections~\ref{sec:mechanisms}--\ref{sec:polynomial-witness} @@ -374,7 +420,7 @@ Under the weak, cut-factorizing hypothesis above, $\widetilde{\mathcal M}_S(\rho \Pi_\lambda^{(S^c)}\,\widetilde{\mathcal M}_S(\rho)\,\Pi_\mu^{(S)}=0 \qquad\text{whenever }\lambda\neq\mu, \end{equation*} -exactly, not merely approximately or in a bound. Moreover, on the surviving diagonal blocks, +exactly, not merely approximately or in a bound. Moreover, whenever the real irreducible $V_\lambda$ is of \emph{real type} (Remark~\ref{rem:real-schur} below) -- the case for every representation actually used in this note -- the surviving diagonal blocks take the Wigner--Eckart form \begin{equation} \widetilde{\mathcal M}_S(\rho)\big|_\lambda = A_\lambda\otimes\mathrm{id}_{V_\lambda} \label{eq:wigner-eckart} @@ -383,9 +429,14 @@ for a unique linear map $A_\lambda:M_\lambda^{(S)}\to M_\lambda^{(S^c)}$, the re \end{proposition} \begin{proof} -$G$-invariance of $\rho$ gives $\mathrm{Ad}^{(S^c)}_g\,\widetilde{\mathcal M}_S(\rho)=\widetilde{\mathcal M}_S(\rho)\,\mathrm{Ad}^{(S)}_g$ for all $g$, i.e.\ $\widetilde{\mathcal M}_S(\rho)$ is a $G$-equivariant map between the two representations~\eqref{eq:isotypic-decomp}. Both claims are then Schur's lemma applied to the isotypic decomposition: equivariant maps vanish between inequivalent irreducible summands, and act as a fixed scalar multiple of the identity on the irreducible factor $V_\lambda$ within a matching pair, leaving exactly the freedom recorded in $A_\lambda$ on the multiplicity spaces. +$G$-invariance of $\rho$ gives $\mathrm{Ad}^{(S^c)}_g\,\widetilde{\mathcal M}_S(\rho)=\widetilde{\mathcal M}_S(\rho)\,\mathrm{Ad}^{(S)}_g$ for all $g$, i.e.\ $\widetilde{\mathcal M}_S(\rho)$ is a $G$-equivariant map between the two representations~\eqref{eq:isotypic-decomp}. The block-vanishing claim is Schur's lemma applied to the isotypic decomposition: an equivariant map between real representations vanishes on isotypic components indexed by inequivalent irreducibles, with no further hypothesis needed. On a matching pair of real-type irreducible factors $V_\lambda$, the same equivariant map, restricted to $V_\lambda\to V_\lambda$, lies in $\mathrm{End}_G(V_\lambda)$, which by hypothesis is $\mathbb R\cdot\id$ (Remark~\ref{rem:real-schur}), giving Eq.~\eqref{eq:wigner-eckart} with $A_\lambda$ recording exactly the remaining freedom on the multiplicity spaces. \end{proof} +\begin{remark}[Real, complex, and quaternionic type: where the real-type restriction is used, and where it is automatic] +\label{rem:real-schur} +Real Schur's lemma states only that $\mathrm{End}_G(V_\lambda)$ is a real division algebra for an irreducible real representation $V_\lambda$; by the Frobenius classification this is $\mathbb R$, $\mathbb C$, or $\mathbb H$, according to whether $V_\lambda$ is of real, complex, or quaternionic type (equivalently, Frobenius--Schur indicator $+1$, $0$, or $-1$). Only in the real-type case does an equivariant self-map of $V_\lambda$ reduce to a \emph{scalar} multiple of the identity as Eq.~\eqref{eq:wigner-eckart} asserts; in the complex or quaternionic case $A_\lambda$ would need to be valued in $\mathrm{Hom}_{\mathbb C}$ or $\mathrm{Hom}_{\mathbb H}$ of the respective multiplicity spaces, and Eq.~\eqref{eq:wigner-eckart} in its literal stated form does not hold. This restriction is genuinely inactive for two of the three symmetry mechanisms used in this note, but not the third, and it is worth being precise about which is which rather than claiming it away uniformly. Every representation appearing in Section~\ref{sec:full-su2-symmetry} is built entirely from $\mathcal V_0^{(a)}\cong\mathbb R^3$, the real (adjoint, spin-$1$) representation of $SO(3)$, and every irreducible integer-spin representation of $SO(3)$ is of real type (realizable over $\mathbb R$, e.g.\ via real spherical harmonics); half-integer spins, where the quaternionic case would arise, never occur in tensor powers of the spin-$1$ generator space, so Eq.~\eqref{eq:wigner-eckart} applies there exactly as stated. The permutation representations of Section~\ref{sec:mechanisms} used for $W_n$ and the ring/graph states are likewise real type (permutation matrices are real orthogonal, and the relevant irreducibles are the standard real irreducibles of $S_n$). The $U(1)$ mechanism of Section~\ref{sec:u1-example}, by contrast, is \emph{not} an instance of the real-type case: the real $2$-dimensional representation of $U(1)$ on the $(x,y)$-plane orthogonal to the rotation axis has $\mathrm{End}_{U(1)}(\mathbb R^2)\cong\mathbb C$ (generated by $\id$ and the rotation-by-$90^\circ$ matrix $J$ with $J^2=-\id$), i.e.\ it is of \emph{complex} type, and Eq.~\eqref{eq:wigner-eckart} does not apply to it in the literal real-scalar form stated above. Section~\ref{sec:u1-example} does not run into this obstruction because it does not use Eq.~\eqref{eq:wigner-eckart} at all for the $U(1)$ case: it complexifies first and works directly with the resulting pair of $1$-dimensional \emph{complex} weight spaces $E_+,E_-$ (mutually conjugate, of opposite charge), which is the standard and correct route for a complex-type real irreducible, and reduces the block structure to the elementary charge-conservation selection rule of Remark~\ref{rem:abelian-selection-rule} instead of a real Wigner-Eckart reduction. So the real-type restriction is inactive throughout Section~\ref{sec:full-su2-symmetry} and the permutation mechanism of Section~\ref{sec:mechanisms}, and is sidestepped rather than violated in Section~\ref{sec:u1-example}. +\end{remark} + Proposition~\ref{prop:block-diagonal} organizes $\widetilde{\mathcal M}_S(\rho)$ by symmetry channel alone, treating $\V_0^{(S)}$ and $\V_0^{(S^c)}$ as undifferentiated representations of $G$. But these spaces already carry the combinatorial sector grading of \cite{aschauer2026a}, $\V_0^{(S)}=\bigoplus_{\emptyset\neq V\subseteq S}\V_V^{(S)}$. The next result shows that this grading and the isotypic one are never in tension: for any symmetry acting locally by conjugation, one refines the other, and both can be imposed simultaneously without contradiction. \begin{proposition}[Compatibility of sector and symmetry decompositions] @@ -483,7 +534,7 @@ By hypothesis $U_\pi\rho U_\pi^\dagger=\rho$, giving Eq.~\eqref{eq:transitive-eq \begin{proposition}[Reduced matrix element formula] \label{prop:reduced-formula} -Let $\rho=T_G(\rho_S\otimes\sigma_{S^c})$ be the $G$-twirl of a product state across $S\mid S^c$, so $\rho$ is separable (a mixture, over $g\in G$, of product states) and $G$-invariant. Let $r^{(S)}\in\V_0^{(S)}$, $v_{S^c}\in\V_0^{(S^c)}$ be the traceless Bloch vectors of $\rho_S$, $\sigma_{S^c}$, with isotypic components $r_\lambda$, $v_\lambda$. Then +Assume $V_\lambda$ is of real type (Remark~\ref{rem:real-schur}; automatic throughout this note). Let $\rho=T_G(\rho_S\otimes\sigma_{S^c})$ be the $G$-twirl of a product state across $S\mid S^c$, so $\rho$ is separable (a mixture, over $g\in G$, of product states) and $G$-invariant. Let $r^{(S)}\in\V_0^{(S)}$, $v_{S^c}\in\V_0^{(S^c)}$ be the traceless Bloch vectors of $\rho_S$, $\sigma_{S^c}$, with isotypic components $r_\lambda$, $v_\lambda$. Then \begin{equation} A_\lambda=\frac{1}{d_\lambda}\,\tilde v_\lambda\,\tilde r_\lambda^{\,T}, \label{eq:reduced-element} @@ -567,20 +618,20 @@ For each two-qubit cluster $S$ and $S^c$, the local weight decomposition $\{-1,0 \toprule $q_S\to q_T$ & block size & singular value(s) & origin \\ \midrule -$+2\to-2$ & $1\times1$ & $\tfrac23\sin(2\varphi)$ & $E_+E_+\to E_-E_-$ \\ +$+2\to-2$ & $1\times1$ & $\tfrac23\lvert\sin(2\varphi)\rvert$ & $E_+E_+\to E_-E_-$ \\ $+1\to-1$ & $2\times2$ & $0,0$ & identically zero for every $\varphi$ \\ $0\to0$ & $3\times3$, rank $1$ & $\tfrac13$ & entirely from $E_0E_0\to E_0E_0$, i.e.\ $ZZ\to ZZ$ \\ $-1\to+1$ & $2\times2$ & $0,0$ & identically zero for every $\varphi$ \\ -$-2\to+2$ & $1\times1$ & $\tfrac23\sin(2\varphi)$ & $E_-E_-\to E_+E_+$ \\ +$-2\to+2$ & $1\times1$ & $\tfrac23\lvert\sin(2\varphi)\rvert$ & $E_-E_-\to E_+E_+$ \\ \bottomrule \end{tabular} \end{center} -so that, exactly, +so that, exactly and for every $\varphi\in\mathbb R$ (not merely $\varphi\in[0,\pi/2]$, where the entries $\tfrac23\sin(2\varphi)$ computed directly from the raw block happen already to be nonnegative), \begin{equation} -\norm{\mathcal M_{AB}(\rho_\varphi)}_* = \frac13+\frac43\sin(2\varphi). +\norm{\mathcal M_{AB}(\rho_\varphi)}_* = \frac13+\frac43\lvert\sin(2\varphi)\rvert. \label{eq:u1-example-formula} \end{equation} -The $q=0$ value $\tfrac13$ is $\varphi$-independent, because $ZZZZ$ is diagonal in the computational basis and both terms of $\lvert\psi_\varphi\rangle$ give it the same eigenvalue $+1$; the entire $\varphi$-dependence, and with it the entanglement signal, sits in the $q=\pm2$ sectors. Equation~\eqref{eq:u1-example-formula} exceeds the cut-separable bound of Theorem~\ref{thm:cluster-cut} exactly when $\sin(2\varphi)>1/2$, i.e.\ for $\varphi\in(\pi/12,\,5\pi/12)$; it vanishes at $\varphi=0$, where $\rho_\varphi$ is itself a product state across $S\mid S^c$, and reaches $\norm{\mathcal M_{AB}(\rho_\varphi)}_*=5/3$ at the maximally superposed point $\varphi=\pi/4$. +The two $q=\pm2$ singular values are genuinely $\lvert\tfrac23\sin(2\varphi)\rvert$, not $\tfrac23\sin(2\varphi)$ itself -- the raw $1\times1$ block entries computed from $\mathcal C(\rho_\varphi)$ can be negative, but a singular value cannot, and the nuclear norm sums the absolute values (verified directly against the true nuclear norm of the full raw $9\times9$ block for $\varphi$ outside $[0,\pi/2]$ in \texttt{scripts/check\_u1.py}). The $q=0$ value $\tfrac13$ is $\varphi$-independent, because $ZZZZ$ is diagonal in the computational basis and both terms of $\lvert\psi_\varphi\rangle$ give it the same eigenvalue $+1$; the entire $\varphi$-dependence, and with it the entanglement signal, sits in the $q=\pm2$ sectors. Equation~\eqref{eq:u1-example-formula} exceeds the cut-separable bound of Theorem~\ref{thm:cluster-cut} exactly when $\lvert\sin(2\varphi)\rvert>1/2$; restricting to $\varphi\in[0,\pi/2]$ for concreteness, this is $\varphi\in(\pi/12,\,5\pi/12)$, it vanishes at $\varphi=0$, where $\rho_\varphi$ is itself a product state across $S\mid S^c$, and reaches $\norm{\mathcal M_{AB}(\rho_\varphi)}_*=5/3$ at the maximally superposed point $\varphi=\pi/4$. This example illustrates Corollary~\ref{cor:joint-refinement} concretely: the full $9\times9$ block factors, without any state-specific input, into one $3\times3$ block of rank $1$, two mutually zero $2\times2$ blocks, and two $1\times1$ blocks, purely by charge conservation --- before any computation is needed to know which entries can possibly be nonzero. Since $\rho_\varphi$ is generically not a stabilizer state, this degeneracy pattern cannot be attributed to Lemma~\ref{lem:stabilizer-degeneracy}, giving a concrete instance of the separation of mechanisms described in general terms by Remark~\ref{rem:two-mechanisms}. @@ -723,25 +774,27 @@ The point of Eq.~\eqref{eq:coherence-decomposition} is that $T_1,T_2,C_{12}$ do \begin{proposition}[Cut-independent template data] \label{prop:coherence-templates} -Let $\{\ket{\phi_a}\}_{a=1}^r$ be finitely many pure states, each an exact eigenvector (not merely a stabilizer of the density matrix) of the collective action~\eqref{eq:collective-hypothesis} for a common $G$, i.e.\ $U_g^{\otimes n}\ket{\phi_a}=\ket{\phi_a}$ for every $g\in G$ and every $a$. Define the $r^2$ fixed tensors $T_{ab}(\vec\imath):=\bra{\phi_a}\sigma_{\vec\imath}\ket{\phi_b}$ (so $T_{aa}=T_a$ of Eq.~\eqref{eq:coherence-decomposition} and $T_{ab}+T_{ba}=C_{ab}$ for $a\ne b$). Then for every $\rho=\sum_{a,b}c_{ab}\ketbra{\phi_a}{\phi_b}$ in the span of $\{\ketbra{\phi_a}{\phi_b}\}$ with $\tr\rho=1$, $\rho\ge0$, the full correlation tensor is +Let $\{\ket{\phi_a}\}_{a=1}^r$ be finitely many pure states, each an exact eigenvector (not merely a stabilizer of the density matrix) of the collective action~\eqref{eq:collective-hypothesis} for a common $G$, i.e.\ $U_g^{\otimes n}\ket{\phi_a}=\ket{\phi_a}$ for every $g\in G$ and every $a$. Define the $r^2$ fixed \emph{full} Bloch tensors $T_{ab}(\vec\imath):=\bra{\phi_a}\sigma_{\vec\imath}\ket{\phi_b}$, now over the \emph{full} index range $\vec\imath\in\{0,1,2,3\}^n$ of \cite{aschauer2026a}'s Definition of $\mathcal C(\rho)$ (i.e.\ $\sigma_0=\id$ is allowed on each leg, not only the all-active sector $\vec\imath\in\{x,y,z\}^n$ used in the worked examples below) -- so $T_{aa}=T_a$ of Eq.~\eqref{eq:coherence-decomposition} and $T_{ab}+T_{ba}=C_{ab}$ for $a\ne b$, now as full tensors. Then for every $\rho=\sum_{a,b}c_{ab}\ketbra{\phi_a}{\phi_b}$ in the span of $\{\ketbra{\phi_a}{\phi_b}\}$ with $\tr\rho=1$, $\rho\ge0$, the full correlation tensor is \begin{equation} -T(\rho) = \sum_{a,b} c_{ab}\,T_{ab}, +\mathcal C(\rho) = \sum_{a,b} c_{ab}\,T_{ab}, \label{eq:general-coherence-decomposition} \end{equation} -independent of which cut's matricization is subsequently taken. +independent of which cut's matricization, or which sector's slice (including the all-active one), is subsequently taken. \end{proposition} \begin{proof} -$T(\rho)(\vec\imath)=\tr(\rho\,\sigma_{\vec\imath})$ 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 $T_{ab}$. +$\mathcal C(\rho)(\vec\imath)=\tr(\rho\,\sigma_{\vec\imath})$ 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 or sector dependence enters only afterward, through reshaping or slicing the full index set $\vec\imath\in\{0,1,2,3\}^n$, and does not affect the coefficients $c_{ab}$ or the tensors $T_{ab}$. \end{proof} \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 $a3$, which would require checking (or proving in closed form) the analogous identity to Eq.~\eqref{eq:xi-sum} for correspondingly larger $p,j$; (ii) a derivation of the conjugation phase Eq.~\eqref{eq:conjugation-phase} for general $n_\ell$ rather than the two values used here, and at exact rather than floating-point precision; (iii) identification of Eq.~\eqref{eq:xi-sum} with a specific named closed-form formula in the classical $6j$-symbol literature (it 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) -- two attempts at matching the exact literature convention (transcribed as comments in \texttt{scripts/search\_6j.py}) failed to reproduce Eq.~\eqref{eq:xi-sum}, and the discrepancy was not tracked down, so this is a presentation and cross-referencing gap rather than a gap in Theorem~\ref{thm:six-j-recoupling}'s correctness. \end{remark} \subsubsection{Combining with permutation symmetry: multiplicity-free channels} @@ -848,37 +901,73 @@ Consequently, for a source cluster $S$ of size $m$ that is invariant under \emph \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} +\ket{Q_3} \;\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 $g\in SU(2)$). +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). Explicitly, identifying $\ket{D_3^k}=\ket{j{=}3/2,M{=}k-3/2}$, Eq.~\eqref{eq:dicke-network-state} is, up to the overall (irrelevant) sign convention, the standard two-spin-$j$ singlet $\sum_{M=-j}^{j}(-1)^{j-M}\ket{j,M}\ket{j,-M}/\sqrt{2j+1}$ at $j=3/2$: the coefficient of $\ket{D_3^k}_{ABC}\ket{D_3^{3-k}}_{DEF}$ is $(-1)^{3/2-(k-3/2)}/2=(-1)^{3-k}/2\propto(-1)^k$, matching Eq.~\eqref{eq:dicke-network-state} exactly. By construction $\ket{Q_3}$ 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: $\lvert\bra{Q_3}U_g^{\otimes6}\ket{Q_3}\rvert=1$ 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 +Restricting the $27\times27$ block $M_{S\to S^c}(\ketbra{Q_3}{Q_3})$ 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 (genuinely \emph{signed}, not merely their nuclear norm) scalar reduced matrix elements \begin{equation} -A_1 = \tfrac13, \qquad A_3 = 2, +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 $10^{-14}$). +satisfying $3\lvert A_1\rvert+7\lvert A_3\rvert=1+14=15=\norm{M_{S\to S^c}(\ketbra{Q_3}{Q_3})}_*$ exactly (Corollary~\ref{cor:projector-norm-formula}), 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} + +\begin{remark}[The pattern persists for $m=4,5$, but has no known closed form] +\label{rem:dicke-network-general-m} +Let $\ket{Q_m}\propto\sum_{k=0}^m(-1)^k\ket{D_m^k}_S\otimes\ket{D_m^{m-k}}_{S^c}$ denote the size-$2m$ generalization of Eq.~\eqref{eq:dicke-network-state} (so $\ket{Q_3}$ above is the $m=3$ case). Repeating the construction of Example~\ref{ex:dicke-network} for $m=4$ ($8$ qubits, $\binom62=15$-dimensional symmetric subspace) and $m=5$ ($10$ qubits, $\binom72=21$-dimensional) gives, exactly (\texttt{scripts/01\_double\_collapse\_table.py}): +\begin{center} +\begin{tabular}{@{}lll@{}} +\toprule +$m$ & signed scalars $A_j$ & top ratio \\ +\midrule +$3$ & $A_3=-2,\ A_1=-\tfrac13$ & $A_3/A_1=6$ \\ +$4$ & $A_4=\tfrac{16}5,\ A_2=\tfrac25,\ A_0=\tfrac15$ & $A_4/A_2=8$ \\ +$5$ & $A_5=-\tfrac{16}3,\ A_3=-\tfrac8{15},\ A_1=-\tfrac15$ & $A_5/A_3=10$ \\ +\bottomrule +\end{tabular} +\end{center} +Two patterns hold exactly across all three cases: the common sign is $(-1)^m$ (so $\ket{Q_3},\ket{Q_5}$ give negative $A_j$ throughout, $\ket{Q_4}$ positive throughout), and the ratio of the top two surviving channels is $A_m/A_{m-2}=2m$ exactly. We record these as verified patterns, not as a proved closed form for $A_j(m)$ at general $j0$ -- equivalently $M:=\mathcal M_S(\rho_0)$ has full rank $\min(N,n)$ and $MM^T$ or $M^TM$ (whichever is smaller) equals $\sigma_0^2\cdot\id$, so that in particular \emph{no} singular value of $M$ is zero -- as is the case for $\GHZ_3$, \cite{aschauer2026a} (there $M\in\R^{15\times3}$ has all three singular values equal to $\sqrt{2/3}$, none zero), then \begin{equation*} \left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0} \norm{\mathcal M_S(\rho(\varepsilon))}_*=\tr(K). \end{equation*} -More generally this is the contribution of the $\sigma_0$-block alone; -non-degenerate singular values contribute their own -$u_{0}^T\mathcal M_S(\delta\rho)v_{0}$ term individually, and the total -derivative is the sum over all blocks. +More generally this is the contribution of the $\sigma_0$-block alone; any further \emph{nonzero, simple} singular value $\sigma_1\neq\sigma_0$ of $M$ (i.e.\ $\sigma_1>0$, unrepeated, hence itself covered by Theorem~\ref{thm:degenerate-svd-perturbation} with $m=1$) contributes its own $u_{1}^T\mathcal M_S(\delta\rho)v_{1}$ term individually, and the total derivative is the sum of all such nonzero-block contributions. A genuinely \emph{zero} singular value of $M$, if one is present, is explicitly excluded from this statement and does \emph{not} contribute a linear term in general: since singular values are nonnegative, a zero singular value typically becomes $\lvert\varepsilon\rvert\cdot\lvert u_0^T\mathcal M_S(\delta\rho)v_0\rvert+O(\varepsilon^2)$ under a generic perturbation, so $\norm{\mathcal M_S(\rho(\varepsilon))}_*$ is generally only directionally, not linearly, differentiable whenever $M$ has a zero singular value -- e.g.\ already for the $2\times2$ example $M(\varepsilon)=\mathrm{diag}(1,\varepsilon)$, where $\norm{M(\varepsilon)}_*=1+\lvert\varepsilon\rvert$ has no two-sided derivative at $\varepsilon=0$ despite $\sigma_0=1$ being simple and nonzero there. This is why the hypothesis above insists on \emph{no} zero singular value being present at all, not merely that $\sigma_0$ is the only nonzero one. \end{corollary} \begin{remark}[Why a compression matrix, not a higher-order tensor] @@ -1019,8 +1104,9 @@ featureless) nor a higher-order tensor, but a single real symmetric $m\times m$ matrix per degenerate block --- an ordinary eigenvalue problem, not an SVD. This is the shadow-map instance of the general fact that unitarily invariant matrix norms such as $\norm{\cdot}_*$ are -differentiable at simple spectra but only \emph{directionally} -differentiable at repeated ones, with the one-sided directional +convex, and hence directionally differentiable, everywhere, but are +generally not \emph{Fr\'echet} differentiable at matrices with repeated +nonzero singular values, with the one-sided directional derivative controlled by exactly this kind of compressed, symmetrized perturbation \cite{lewissendov2005}. Two mechanisms of this note make degenerate blocks the rule rather than the exception: @@ -1245,9 +1331,9 @@ 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 +permutationally invariant states \cite{novomoroderguhne2013}, with dedicated closed-form criteria available for Dicke states in particular -\cite{bergmanngühne2013}. All of these results work in the coupled-spin +\cite{bergmannguhne2013}. 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 @@ -1482,20 +1568,27 @@ 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. +originally exploited only the $U(1)$-type or full-$S_m$ permutation +symmetry of a single cluster's own parties in isolation; the extension +to the \emph{full}, non-abelian $SU(2)$ hypothesis, and its combination +with the $S_m$-permutation collapse into a multiplicity-free reduction, +is now carried out in Section~\ref{sec:full-su2-symmetry} +(Sections~\ref{sec:six-j-recoupling}--\ref{sec:sm-so3-combination}). +Three narrower gaps remain within that section specifically: the +recoupling formula of Theorem~\ref{thm:six-j-recoupling} is verified +exactly for $p,j\le3$ rather than proved in closed form for arbitrary +$p,j$ (Remark~\ref{rem:six-j-scope}); the conjugation-phase identity +Eq.~\eqref{eq:conjugation-phase} is likewise verified rather than +derived inductively for general cluster size; and the qudit and +multi-cluster generalizations noted in Remark~\ref{rem:qudit-collapse} +have not been revisited under the combined $S_m\times SU(2)$ hypothesis. +Separately, and still fully open: using the same symmetry constraints +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 this note into a constructive search tool rather than +only an explanatory or verification one, and nothing here attempts +this. Finally, a broader positioning remark. This note and its companion \cite{aschauer2026a} study the singular-value structure of one