Exact partially symmetric rank of products of generalized W tensors
For $d \ge 3$ let $$W_d = \sum_{j=1}^{d} e_1^{\otimes (j-1)} \otimes e_2 \otimes e_1^{\otimes (d-j)} \in \operatorname{Sym}^d(\mathbb{C}^2)$$ be the W-state, which corresponds to the binary form $x^{d-1}y$. For $k \ge 2$ and $d_1, \dots, d_k \ge 3$ let $T_{\mathbf d} = W_{d_1} \otimes \cdots \otimes W_{d_k}$. Its partially symmetric rank is the least $s$ such that $$T_{\mathbf d} = \sum_{\ell=1}^{s} v_{1,\ell}^{\otimes d_1} \otimes \cdots \otimes v_{k,\ell}^{\otimes d_k}, \qquad v_{j,\ell} \in \mathbb{C}^2.$$ Ballico, Bernardi, Christandl and Gesmundo showed that the partially symmetric rank of $W_3 \otimes W_3$ is $8$, and that the rank of $T_{\mathbf d}$ as a tensor in $(\mathbb{C}^2)^{\otimes (d_1 + \cdots + d_k)}$, which is at most its partially symmetric rank, is at least $d_1 + \cdots + d_k - k + 1$. Canino, Casarotti and Santarsiero showed that the partially symmetric rank of $T_{\mathbf d}$ is at most $2^{k-1}(d_1 + \cdots + d_k - 2k + 2)$.
Problem. Determine the partially symmetric rank of $T_{\mathbf d}$ for every $k \ge 2$ and all $d_1, \dots, d_k \ge 3$.
Source: Oneto and Ventura (2025), Question 12; the range k ≥ 2 is OpenProblemsInNLA's reading.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- OpenProblemsInNLA, partially symmetric ranks, W-states
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{oneto2025,
author = {Oneto, Alessandro and Ventura, Emanuele},
title = {Ranks of tensors: geometry and applications},
journal = {Boll. Unione Mat. Ital.},
volume = {18},
number = {3},
year = {2025},
pages = {751--778},
doi = {10.1007/s40574-025-00472-9},
}
@article{ballico2019,
author = {Ballico, Edoardo and Bernardi, Alessandra and Christandl, Matthias and Gesmundo, Fulvio},
title = {On the partially symmetric rank of tensor products of {$W$}-states and other symmetric tensors},
journal = {Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.},
volume = {30},
number = {1},
year = {2019},
pages = {93--124},
doi = {10.4171/RLM/837},
eprint = {1803.01623},
archivePrefix = {arXiv},
}
@article{canino2026,
author = {Canino, Stefano and Casarotti, Alex and Santarsiero, Pierpaola},
title = {A new bound on the rank of tensor product of {$W$}-states},
journal = {Proc. Amer. Math. Soc.},
volume = {154},
number = {12},
year = {2026},
pages = {5095--5117},
doi = {10.1090/proc/17856},
eprint = {2512.05828},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr29,
author = {Townsend, Alex},
title = {{TR-29}: Exact partially symmetric rank of products of generalized W tensors},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-29/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Resolution claims
For $d_1,d_2\ge 2$ let $W_d=x^{d-1}y\in\mathrm{Sym}^d(\mathbb C^2)$ (up to scale) be the W-state. We prove that the partially symmetric rank of $W_{d_1}\otimes W_{d_2}$ equals $2(d_1+d_2-2)$, so the known upper bound of Gałązka and of Canino–Casarotti–Santarsiero is sharp for every pair. This is the two-factor case of Question 12 of Oneto–Ventura. The lower bound combines two elementary facts: a point set whose …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-09 | Open → Partially resolved | The registry's curators | The partially symmetric rank of a product of two W-states and of bihomogeneous binary monomials v1 recorded from OpenProblemsInNLA @ eb7c9d6, manuscript/TR29_two_factor_rank.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00137, "Exact partially symmetric rank of products of generalized W tensors", clopen.solutions.