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bibkey: sutter2026grouptheoretic authors: Tobias C. Sutter, Christopher Popp and Beatrix C. Hiesmayr year: 2026 title: “Group-Theoretic Perspective on the PPT and Realignment Criteria in the Magic Simplex for Bipartite Qutrits” doi: 10.1103/z7jn-4try url: https://arxiv.org/abs/2508.18393v2 claim: “Section VI states as open whether an entanglement criterion based on the entry-wise 1-norm of the Weyl–Bloch vector can be derived for general bipartite qudits, and reports numerically that separable pure two-qutrit states seem to satisfy ‖β‖₁ ≤ 25 while pure states reach ≈ 25.9735.” strata_touched:

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DOI: 10.1103/z7jn-4try

Source: https://arxiv.org/abs/2508.18393v2

Published in Phys. Rev. A 113, 032440 (2026). Version 2 of the arXiv record (24 March 2026; v1 25 August 2025, titled A Group-Theoretic Perspective on the PPT and Realignment Entanglement Criteria) is the quoted version (main.tex); page numbers refer to the v2 PDF. Separability: Section II, p. 2. Weyl operators, eq. (7): p. 3. Bloch representation, eq. (16): p. 5. Entry-wise 1-norm: p. 6. The open problem and the numerical observation: Section VI, p. 10.

Separability (p. 2):

A state $\rho \in \mathcal{D}(\mathcal{H})$ is called separable if it can be written as $\rho ={\sum }_i{p}_i{\rho }_i^A\otimes {\rho }_i^B$ for some set $\{p_i,{\rho }_i^A,{\rho }_i^B\}$, where $p_i\ge 0$ and ${\rho }_i^{A/B}\in \mathcal{D}({\mathcal{H}}_{A/B})$ for all $i$, and ${\sum }_i{p}_i=1$; otherwise, $\rho$ is called entangled.

Weyl–Heisenberg operators, eq. (7) (p. 3):

$W_{k,l}={\sum }_{j=0}^{d-1}{\omega }^{jk}|j⟩⟨j+l|$ where $k,l\in \{0,1,\dots ,d-1\}$ and $\omega :=e^{\frac{2\pi i}{d}}$, and, throughout this paper, variables taking values in $\{0,1,\dots ,d-1\}$ are understood as elements of ${\mathbb{Z}}_d$.

Bloch representation, eq. (16) (p. 5):

any bipartite qudit state can be written as $\rho =\frac{1}{d^2}{\sum }_{i,j,k,l=0}^{d-1}{\beta }_{ij,kl}{W}_{i,j}\otimes W_{k,l}$ … Using ${\beta }_{ij,kl}=\mathrm{tr}\left(\rho (W_{i,j}\otimes W_{k,l}{)}^{†}\right)$ …

The open problem (Section VI, p. 10):

Going forward, it is furthermore an open problem whether for general bipartite qudit states, an entanglement criterion based on the entry-wise 1-norm of the Bloch vector, similar to Thm. IV.1, can be derived. Ref. [52, 53] present a sufficient criterion for separability, reading $\Vert \beta {\Vert }_1\le 2⟹\rho \in \mathrm{SEP}$ for $\beta$ as in (16), but to the authors’ knowledge, no corresponding sufficient entanglement criterion exists. A naive approximation using $\Vert \beta {\Vert }_1\le d^2\Vert \beta {\Vert }_2$ for $\beta \in {\mathbb{C}}^{d^4}$ yields that $\rho$ is entangled if $\Vert \beta {\Vert }_1>d^3$. However, the maximal value for pure states $|\psi ⟩\in {\mathbb{C}}^d\otimes {\mathbb{C}}^d$ is also $\Vert \beta {\Vert }_1=d^3$. A quick numerical optimization for $d=3$ yields $\Vert \beta {\Vert }_1\lesssim 25.9735$ for general pure states $|\psi ⟩\in {\mathbb{C}}^3\otimes {\mathbb{C}}^3$, whereas separable pure states seem to satisfy $\Vert \beta {\Vert }_1\le 25$.