diff --git a/paper/references.bib b/paper/references.bib index 39dafd3..3b971e0 100644 --- a/paper/references.bib +++ b/paper/references.bib @@ -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.} +} diff --git a/paper/symmetric_shadow_maps_formal.tex b/paper/symmetric_shadow_maps_formal.tex index f01035b..96ca525 100644 --- a/paper/symmetric_shadow_maps_formal.tex +++ b/paper/symmetric_shadow_maps_formal.tex @@ -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} % ============================================================