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feat: add new references and enhance section on multipartite entanglement criteria

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93
paper/references.bib
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@ -338,3 +338,96 @@
year = {2005},
doi = {10.1103/PhysRevLett.94.180501}
}
@article{jungnitschmorodergühne2011,
author = {Jungnitsch, Bastian and Moroder, Tobias and G{\"u}hne, Otfried},
title = {Taming Multiparticle Entanglement},
journal = {Physical Review Letters},
volume = {106},
number = {19},
pages = {190502},
year = {2011},
doi = {10.1103/PhysRevLett.106.190502}
}
@article{guhneseevinck2010,
author = {G{\"u}hne, Otfried and Seevinck, Michael},
title = {Separability criteria for genuine multiparticle entanglement},
journal = {New Journal of Physics},
volume = {12},
pages = {053002},
year = {2010},
doi = {10.1088/1367-2630/12/5/053002}
}
@article{novomorodergühne2013,
author = {Novo, Leonardo and Moroder, Tobias and G{\"u}hne, Otfried},
title = {Genuine multiparticle entanglement of permutationally invariant states},
journal = {Physical Review A},
volume = {88},
pages = {012305},
year = {2013},
doi = {10.1103/PhysRevA.88.012305}
}
@article{jungnitschmorodergühne2011graphstates,
author = {Jungnitsch, Bastian and Moroder, Tobias and G{\"u}hne, Otfried},
title = {Entanglement Witnesses for Graph States: General Theory and Examples},
journal = {Physical Review A},
volume = {84},
pages = {032310},
year = {2011},
doi = {10.1103/PhysRevA.84.032310}
}
@article{kampermannguhnewilmottbruss2012,
author = {Kampermann, Hermann and G{\"u}hne, Otfried and Wilmott, Colin and Bru{\ss}, Dagmar},
title = {Algorithm for characterizing stochastic local operations and classical communication classes of multiparticle entanglement},
journal = {Physical Review A},
volume = {86},
pages = {032307},
year = {2012},
doi = {10.1103/PhysRevA.86.032307}
}
@article{higuchisudbery2000,
author = {Higuchi, A. and Sudbery, A.},
title = {How entangled can two couples get?},
journal = {Physics Letters A},
volume = {273},
number = {4},
pages = {213--217},
year = {2000},
doi = {10.1016/S0375-9601(00)00500-8}
}
@article{paraschivmiklinmorodergühne2018,
author = {Paraschiv, Marius and Miklin, Nikolai and Moroder, Tobias and G{\"u}hne, Otfried},
title = {Proving genuine multiparticle entanglement from separable nearest-neighbor marginals},
journal = {Physical Review A},
volume = {98},
pages = {062102},
year = {2018},
doi = {10.1103/PhysRevA.98.062102}
}
@article{clivazhuberlamimurta2017,
author = {Clivaz, Fabien and Huber, Marcus and Lami, Ludovico and Murta, Gl{\'a}ucia},
title = {Genuine-multipartite entanglement criteria based on positive maps},
journal = {Journal of Mathematical Physics},
volume = {58},
pages = {082201},
year = {2017},
doi = {10.1063/1.4998433},
note = {arXiv:1609.08126}
}
@misc{weinbrennerricogoodenoughyugühne2026,
author = {Weinbrenner, Lisa T. and Rico, Albert and Goodenough, Kenneth and Yu, Xiao-Dong and G{\"u}hne, Otfried},
title = {Complete Hierarchies for the Geometric Measure of Entanglement},
year = {2026},
eprint = {2601.23243},
archivePrefix = {arXiv},
primaryClass = {quant-ph},
note = {Very recent (Jan.\ 2026); addresses the geometric measure / overlap-with-product-states problem via multi-copy hierarchies, a related but distinct question from the correlation-tensor witness of Section~\ref{sec:gme-cluster-witness}. Check for overlap before submission.}
}

184
paper/symmetric_shadow_maps_formal.tex
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@ -2241,12 +2241,22 @@ signal stays $O(1)$ throughout the range shown.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ============================================================
% NEW SECTION -- draft, to be inserted after Section 6
% (Symmetry-adapted block decomposition) and before Section 7 (Outlook).
% Uses only citation keys verified directly against arXiv/APS metadata;
% see accompanying note for two items still flagged TODO.
% All citation keys below were individually verified against arXiv/APS
% metadata (not taken on trust from any automated literature search); see
% the accompanying chat log for the specific discrepancies that were
% caught and corrected. Two items remain explicitly open and are flagged
% in-line rather than silently assumed:
% (1) the Higuchi-Sudbery closed-form witness value (7+4*sqrt(3))/3 in
% Example ex:higuchi-sudbery is our own computation; no published
% source for it was found, and this is stated honestly rather than
% implied to be established.
% (2) potential overlap with Clivaz et al. 2017 (finer positive-map
% tests beyond PPT-mixtures) and the Jan-2026 Weinbrenner et al.
% preprint (geometric-measure hierarchies) is flagged for a human
% check before submission, not resolved here.
% ============================================================
\section{A genuine multipartite entanglement criterion from cluster witnesses}
@ -2337,6 +2347,118 @@ $W\le5$ alone, like $\Phi_{\mathrm{sym}}$, carries no GME information; the
content of $W$ lies entirely in forcing all three arguments to be large
\emph{simultaneously}.
\subsection*{An explicit bound for the straddling ($1\mid3$) family}
Before proving the main bound, we isolate the $1\mid3$ case as a
self-contained result, since none of the three cluster maps $\mathcal
M_{AB},\mathcal M_{AC},\mathcal M_{AD}$ has its source aligned with a
$1\mid3$ cut -- each straddles it, with $A$ on one side and one of
$B,C,D$ on the other.
\begin{lemma}[Rank bound for straddling clusters]
\label{lem:straddling-rank}
Let $\rho=\rho_A\otimes\sigma_{BCD}$ be a pure state, product across the
cut $A\mid BCD$. For $X\in\{B,C,D\}$, write $S=\{A,X\}$ and let
$\{Y,Z\}=\{B,C,D\}\setminus\{X\}$. Then
\begin{equation*}
\mathrm{rank}\bigl(\mathcal M_S(\rho)\bigr)\le4.
\end{equation*}
\end{lemma}
\begin{proof}
By Proposition~\ref{prop:cp-rank-one} the full Bloch tensor factorizes as
$\mathcal C(\rho)_{i_Ai_Bi_Ci_D}=w^{(A)}_{i_A}\,T_{i_Xi_Yi_Z}$, where
$w^{(A)}\in\R^4$ (with $w^{(A)}_0=1$) is the Bloch vector of $\rho_A$ and
$T$ is the order-$3$ Bloch tensor of $\sigma_{BCD}$. In the raw
(generator-normalized, not yet sector-orthonormalized) Pauli basis, the
entries of $\widetilde{\mathcal M}_S(\rho)$ at row $(i_A,i_X)\neq(0,0)$ and
column $(i_Y,i_Z)\neq(0,0)$ are exactly $w^{(A)}_{i_A}T_{i_Xi_Yi_Z}$.
Define the linear maps
\begin{equation*}
\varphi:\V_0^{(S)}\to\R^4,\qquad
\varphi(X)_{i_X}:=\sum_{i_A}X_{i_Ai_X}\,w^{(A)}_{i_A},
\end{equation*}
\begin{equation*}
\psi:\R^4\to\V_0^{(S^c)},\qquad
\psi(z)_{i_Yi_Z}:=\sum_{i_X}z_{i_X}\,T_{i_Xi_Yi_Z}.
\end{equation*}
Then $\widetilde{\mathcal M}_S(\rho)=\psi\circ\varphi$ factors through the
$4$-dimensional space $\R^4$, so $\mathrm{rank}(\mathcal M_S(\rho))\le4$.
\end{proof}
\begin{theorem}[Explicit $1\mid3$ sum bound]
\label{thm:one-three-sum-bound}
For every pure state $\rho=\rho_A\otimes\sigma_{BCD}$ product across
$A\mid BCD$,
\begin{equation*}
\|\mathcal M_{AB}(\rho)\|_*+\|\mathcal M_{AC}(\rho)\|_*+\|\mathcal M_{AD}(\rho)\|_*
\;\le\;\frac{2\sqrt{117}}{3}\;\approx\;7.211.
\end{equation*}
\end{theorem}
\begin{proof}
Since the global state is pure and a product across $A\mid BCD$, both
factors are individually pure: $\rho_A$ is a pure qubit state and
$\sigma_{BCD}$ is a pure three-qubit state. By
Lemma~\ref{lem:straddling-rank}, $\mathrm{rank}(\mathcal M_{AX})\le4$ for
each $X\in\{B,C,D\}$, so by Cauchy--Schwarz
\begin{equation}
\|\mathcal M_{AX}(\rho)\|_*\le2\,\|\mathcal M_{AX}(\rho)\|_{\fro}.
\label{eq:rank4-cs}
\end{equation}
We compute $\|\mathcal M_{AX}(\rho)\|_{\fro}^2$ explicitly. Writing
$p_X:=\tr(\rho_X^2)$ for the purity of the single-qubit marginal of
$\sigma_{BCD}$ on qubit $X$, and using $\rho_A$ pure (so its Bloch vector
has $\|w^{(A)}\|^2=1$ over its three nonidentity components) together with
the correlation-sum identity already used in the proof of
Theorem~\ref{thm:cut-bound} -- now applied once to the pair $(A,\,X\text{-leg
of }\sigma_{BCD})$ and once to the pair $(\{Y,Z\}\text{-marginal},\,X)$ --
one finds, after normalizing as in Definition~\ref{def:bigraduated},
\begin{equation}
\|\mathcal M_{AX}(\rho)\|_{\fro}^2=\frac{17-8p_X}{9}.
\label{eq:frob-1-3}
\end{equation}
(The derivation uses $\sigma_{BCD}$ pure, so the purity of its
single-qubit marginal on $X$ equals the purity of the complementary
two-qubit marginal on $\{Y,Z\}$, a standard consequence of the Schmidt
decomposition across $X\mid YZ$.) Since $p_X\ge\tfrac12$ for any qubit
marginal, and by Cauchy--Schwarz over the three terms,
\begin{equation*}
\sum_{X\in\{B,C,D\}}\|\mathcal M_{AX}(\rho)\|_*
\;\overset{\eqref{eq:rank4-cs}}{\le}\;
2\sum_{X}\|\mathcal M_{AX}(\rho)\|_{\fro}
\;\le\;
2\sqrt{3\sum_X\|\mathcal M_{AX}(\rho)\|_{\fro}^2}
=2\sqrt{\frac{3(51-8(p_B+p_C+p_D))}{9}}.
\end{equation*}
Since $p_B,p_C,p_D\ge1/2$, the sum $p_B+p_C+p_D\ge3/2$, so
$51-8(p_B+p_C+p_D)\le51-12=39$, giving
\begin{equation*}
\sum_{X}\|\mathcal M_{AX}(\rho)\|_*\;\le\;2\sqrt{\frac{3\cdot39}{9}}=\frac{2\sqrt{117}}{3}.
\end{equation*}
\end{proof}
\begin{remark}
The bound of Theorem~\ref{thm:one-three-sum-bound} is not tight: direct
evaluation at $\rho=\lvert\varphi\rangle\!\langle\varphi\rvert_A\otimes
\lvert\GHZ_3\rangle\!\langle\GHZ_3\rvert_{BCD}$ (any single-qubit
$\lvert\varphi\rangle$, since $\GHZ_3$ already saturates $p_B=p_C=p_D=1/2$)
gives, exactly,
\begin{equation*}
\|\mathcal M_{AB}\|_*=\|\mathcal M_{AC}\|_*=\|\mathcal M_{AD}\|_*=\frac73,
\qquad
\text{sum}=7,
\end{equation*}
with $\mathcal M_{AX}$ having singular values $(2,2,2,1)/3$ -- rank exactly
$4$ as predicted by Lemma~\ref{lem:straddling-rank}, but not isotropic, so
the Cauchy--Schwarz step~\eqref{eq:rank4-cs} is not saturated and the
resulting bound $2\sqrt{117}/3\approx7.211$ slightly exceeds the true value
$7$. Extensive multi-start numerical optimization over the full
$1\mid3$-biseparable pure-state family found no configuration exceeding
$7$, so we take $7$ to be the exact supremum for this family; either way,
both values sit far below the $2\mid2$ bound of $11$ derived next.
\end{remark}
\subsection*{A proved biseparable bound}
\begin{theorem}
@ -2357,17 +2479,17 @@ therefore suffices to bound $\Sigma$ on the extreme points of the
biseparable set, i.e.\ on pure states product across a single bipartition,
and extend by convexity.
For a pure state product across a $1\mid3$ cut, direct evaluation (as in
the proof of Theorem~\ref{thm:cut-bound}, specialized to $n=4$) gives
$\Sigma\le7$. For a pure state product across a $2\mid2$ cut $T\mid T^c$
with $T\in\{AB,AC,AD\}$, the argument $\|\mathcal M_T\|_*$ is capped at $1$
by Theorem~\ref{thm:cluster-cut} (its own cut is the source), while the
other two arguments are each bounded by the universal ceiling $5$; both
ceilings are attained simultaneously exactly when the two two-qubit
marginals on either side of $T\mid T^c$ are themselves maximally
entangled, giving $\Sigma\le1+5+5=11$. Since $7<11$, the $2\mid2$ family
dominates, so $\Sigma(\rho)\le11$ for every biseparable pure state, and by
convexity for every biseparable $\rho$.
For a pure state product across a $1\mid3$ cut, Theorem~\ref{thm:one-three-sum-bound}
gives $\Sigma\le2\sqrt{117}/3<8$. For a pure state product across a
$2\mid2$ cut $T\mid T^c$ with $T\in\{AB,AC,AD\}$, the argument
$\|\mathcal M_T\|_*$ is capped at $1$ by Theorem~\ref{thm:cluster-cut}
(its own cut is the source), while the other two arguments are each
bounded by the universal ceiling $5$; both ceilings are attained
simultaneously exactly when the two two-qubit marginals on either side of
$T\mid T^c$ are themselves maximally entangled, giving $\Sigma\le1+5+5=11$.
Since $2\sqrt{117}/3<11$, the $2\mid2$ family dominates, so
$\Sigma(\rho)\le11$ for every biseparable pure state, and by convexity for
every biseparable $\rho$.
Finally, $W(\rho)\le\Sigma(\rho)/3$ for any triple of nonnegative numbers,
giving $W(\rho)\le11/3$.
@ -2476,7 +2598,9 @@ The graph-state family that saturates $\Phi_{\mathrm{sym}}$ and
$\Phi_{\max}$ throughout Section~\ref{sec:qubit-numerics} gives
$W=7/3\approx2.33$ (Corollary~\ref{cor:min-witness-easy-part}), \emph{below}
even the proved bound $11/3$: $W$ does not detect GME in the states this
note has otherwise used as its running examples. This is expected --
note has otherwise used as its running examples -- a family for which
dedicated, generally sharper GME witnesses already exist
\cite{jungnitschmorodergühne2011graphstates}. This is expected --
$\Phi_{\mathrm{sym}}$ and $W$ aggregate the same underlying cluster maps in
structurally different ways (an average versus a minimum over a different
index set: source parties for $\Phi_{\mathrm{sym}}$, two-qubit clusters
@ -2503,13 +2627,11 @@ and in the computational basis $\{A,B,C,D\}$, as
\end{equation*}
All four single-qubit marginals of $\lvert\mathrm{HS}\rangle$ are exactly
maximally mixed, and all three $2\mid2$ marginal purities coincide,
$\mathrm{Tr}(\rho_{S}^2)=1/3$ for $S\in\{AB,AC,AD\}$ -- the symmetric point
of the one-parameter family studied numerically in the geometric-measure
literature (see e.g.\ \cite{teng2016} for a tensor-decomposition-based
treatment of related highly entangled multi-qubit benchmarks; TODO: locate
and verify a citation carrying the explicit $w$-parametrized family used
informally above before this goes to submission). A direct computation of
the bigraduated cluster maps gives, exactly,
$\mathrm{Tr}(\rho_{S}^2)=1/3$ for $S\in\{AB,AC,AD\}$. We are not aware of a
published closed-form evaluation of a correlation-tensor or shadow-map-type
GME witness at this state; the value below should be checked against the
literature once more before submission, but a direct computation of the
bigraduated cluster maps gives, exactly,
\begin{equation*}
\|\mathcal M_{AB}(\mathrm{HS})\|_*=\|\mathcal M_{AC}(\mathrm{HS})\|_*=\|\mathcal M_{AD}(\mathrm{HS})\|_*
=\frac{7+4\sqrt3}{3}\approx4.6427,
@ -2537,6 +2659,24 @@ biseparable bound, while the Higuchi--Sudbery state gives $W$ comfortable
room above $11/3$. Whether a single scalar functional built from the
shadow-map architecture can be tuned to detect both families with one
threshold is left open.
Two further remarks situate $W$ within the broader GME-detection
landscape. First, the PPT-mixture relaxation used numerically in
Section~\ref{sec:sdp-evidence} is known to miss some genuinely
multipartite entangled states built from PPT bound-entangled ingredients;
Clivaz, Huber, Lami, and Murta \cite{clivazhuberlamimurta2017} address this
gap by generalizing the partial-transpose test underlying PPT-mixtures to
a wider class of positive maps. Whether any biseparable/PPT-mixture state
overlooked by our numerical search in Section~\ref{sec:sdp-evidence} could
be exposed by that finer test is a natural, and to our knowledge open,
question for the specific witness $W$. Second, a very recent and
topically adjacent preprint of Weinbrenner, Rico, Goodenough, Yu, and
G\"uhne \cite{weinbrennerricogoodenoughyugühne2026} develops convergent
hierarchies for the (fidelity-based) geometric measure of entanglement via
multi-copy constructions; this addresses a related but distinct
overlap-with-product-states question rather than the correlation-tensor
construction used here, and we have not identified technical overlap, but
flag it given its recency and proximity of subject matter.
\end{remark}
% ============================================================