bibkey: huynhvu2024universal authors: Khoi-Nguyen Huynh-Vu; Lin Htoo Zaw; Valerio Scarani year: 2024 title: “Certification of genuine multipartite entanglement in spin ensembles with measurements of total angular momentum” doi: 10.1103/PhysRevA.109.042402 url: https://arxiv.org/abs/2311.00806v2 claim: “Conjecture 3: Consider a spin ensemble. Perform the precession protocol with odd K ≥ 3 on the total angular momentum of the system. If the score P_K > 𝐏_K^conj is obtained, then the spin ensemble is GME.” strata_touched:

• D5/S3/Quantum/Entanglement/PrecessionUniversalThresholdRefutation license: citation-only triage: anchor

Verified locator

DOI: 10.1103/PhysRevA.109.042402

Source: https://arxiv.org/abs/2311.00806v2

Conjecture 3, Result 4 and its piecewise threshold: PDF p. 9. Result 4 asserts certification for odd 3 ≤ K ≤ 21, total constituent spin at most 15, and score greater than the threshold plus 10^{−11}. The precession protocol, Eqs. (1)–(3): PDF p. 2. The definitions of separability and GME: PDF p. 3.

Universal precession threshold

Conjecture 3 states:

Consider a spin ensemble. Perform the precession protocol with odd $K\ge 3$ on the total angular momentum of the system. If the score P_K > \mathbf{P}_K^{\text{\normalfont{conj}}} is obtained, then the spin ensemble is GME.

The score is $P_K=\mathrm{tr}(\rho Q_K)$, with $Q_K=K^{-1}{\sum }_{k=0}^{K-1}\mathrm{pos}(J_k)$ and $J_k=\cos (2\pi k/K)J_x+\sin (2\pi k/K)J_y$. The function pos assigns weight one to positive eigenvalues, one half to zero eigenvalues, and zero to negative eigenvalues.

The source defines the state notions by:

With these notations, a state ${\rho }_{\mathbf{J},{\mathbf{J}}^{\complement }}$ of a spin ensemble is separable over the $\mathbf{J}$-${\mathbf{J}}^{\complement }$ bipartition if ${\rho }_{\mathbf{J},{\mathbf{J}}^{\complement }}={\sum }_k{p}_k{\rho }_{\mathbf{J},k}\otimes {\rho }_{{\mathbf{J}}^{\complement },k}$, where ${\rho }_{\mathbf{J},k}$ (or ${\rho }_{{\mathbf{J}}^{\complement },k}$) is a state within the subspace ${⨂}_{j\in \mathbf{J}}{\mathcal{H}}^{(j)}$ (or ${⨂}_{j^{\prime }\in {\mathbf{J}}^{\complement }}{\mathcal{H}}^{(j^{\prime })}$). Conversely, ${\rho }_{\text{GME}}$ is GME if it is not a convex combination of states separable over any bipartition $\mathbf{J}$: that is, ${\rho }_{\text{GME}}\ne {\sum }_{\mathbf{J}}{p}_{\mathbf{J}}{\rho }_{\mathbf{J},{\mathbf{J}}^{\complement }}$.

The separate note huynhvu2024precession concerns Conjecture 2, the $\{1,K/2\}$ separable bound, from the same paper.