bibkey: chakrabartykundu2026network
authors: Sudip Chakrabarty and Amit Kundu
year: 2026
title: Network nonlocality with tripartite entangled sources
doi: 10.48550/arXiv.2610.35838
url: https://arxiv.org/abs/2610.35838v1
claim: Proposition 1 expresses the squared Bell-state-measurement network optimum through orthogonal-direction quadratic optimization over two complete orthonormal frames and all four central-observable indices; the W-state paragraph gives the lower bound sqrt(13)/3 and reports no larger numerical value.
strata_touched:
  - D5/S3/Quantum/MeasurementGeometry/WStateBellMeasurementOptimum
license: citation-only
triage: anchor

W-state Bell-state-measurement optimum

Verified locator

DOI: 10.48550/arXiv.2610.35838

Source: https://arxiv.org/abs/2610.35838v1

Proposition 1 is on PDF page 4. The W-state paragraph and the table caption are on PDF page 5. Appendix B, beginning on PDF page 11, proves the variational formula of Proposition 1.

Source statement

The W-state paragraph states:

For the $W$ state, ${\mathbf{\tau }}_{33}=-\hat{\mathbf{z}}$ and ${\mathbf{\tau }}_{13}=\frac{2}{3}\hat{\mathbf{x}}$ are orthogonal, which gives ${\mathcal{N}}_c\ge \sqrt{13}/3\approx 1.202$. The numerical optimization over frames finds no larger value.

The table caption states:

For ${\mathcal{N}}_c(W)$ the value $\sqrt{13}/3$ is a proven lower bound, and the numerics find no larger value.

On PDF page 4 the observable sentence is:

We write $A_i={\mathbf{e}}_i\cdot \mathbf{\sigma }$ and $C_j={\mathbf{f}}_j\cdot \mathbf{\sigma }$, and set $A_0=C_0=\mathbb{I}$.

The correlation-vector equation on that page is ${\mathbf{\tau }}_{\mu \nu }=(\mathrm{Tr}[\rho A_{\mu }\otimes {\sigma }_k\otimes C_{\nu }]{)}_{k=1,2,3}$. The next paragraph states:

The network value therefore depends on how much of ${\mathbf{\tau }}_{\mu \nu }$ and ${\mathbf{\tau }}_{{\mu }^{\prime }{\nu }^{\prime }}$ two perpendicular directions can capture. For two vectors $\mathbf{\tau },{\mathbf{\tau }}^{\prime }$ this is

and defines $\Lambda (\mathbf{\tau },{\mathbf{\tau }}^{\prime })={\max }_{\mathbf{n}\perp {\mathbf{n}}^{\prime }}[(\mathbf{\tau }\cdot \mathbf{n}{)}^2+({\mathbf{\tau }}^{\prime }\cdot {\mathbf{n}}^{\prime }{)}^2]$. Proposition 1 states:

For two copies of a three-qubit state $\rho$, Bell-state measurements at the central nodes and projective measurements at the end nodes, the largest value of $\sqrt{|I|}+\sqrt{|J|}$ is ${\mathcal{N}}_c(\rho )$, with

${\mathcal{N}}_c(\rho {)}^2={\max }_{\{{\mathbf{e}}_i\},\{{\mathbf{f}}_j\}}{\max }_{\mu \nu ,{\mu }^{\prime }{\nu }^{\prime }}\Lambda ({\mathbf{\tau }}_{\mu \nu },{\mathbf{\tau }}_{{\mu }^{\prime }{\nu }^{\prime }})$.

The maxima run over the frames of the two central nodes and over the indices $\mu ,\nu ,{\mu }^{\prime },{\nu }^{\prime }\in \{0,1,2,3\}$.

Mathematical scope

The associated Lean statement uses the literal normalized W ket on $(\mathrm{Fin}2\times \mathrm{Fin}2)\times \mathrm{Fin}2$, its rank-one density matrix, and the displayed variational supremum. The conclusion is the exact value $\sqrt{13}/3$ for this Bell-measurement class, including zero and repeated central indices. The larger product-projective measurement class and noisy states are separate questions.

The decisive identity is $9\Vert T(a,c){\Vert }^2=(2a_0{c}_0+2a_1{c}_1-a_2{c}_2{)}^2+4a_2^2+4c_2^2$ for unit vectors $a,c$. Parseval identities for complete orthonormal frames bound every index pair; the standard frame attains $13/9$ before taking the square root. This is a repository derivation of the upper bound, rather than an attribution of that upper bound to the source.

Evidence boundary

The preregistration is issue 23251. It identifies the exact experiment in the-omega-institute/trureturing-experiments, report commit 3e3d376e925060a7fa7e3a299ae6e2f7b98edfa0, path docs/reports/wbsm-2610/, integrated branch tip 5ec051dd6edc3b3bafa06423169bd4da1c5d382d, checker SHA-256 aac9922feda7ca6dc2a991f15dcfce11a266be5d9b4df051a9c92d89ddb317a2, reported exit code 0 and final line ALL_CHECKS_OK. Those finite checks are supplementary evidence and do not establish the universal optimum. The literature screen in that issue is ASSUMED-UNVERIFIED for completeness.