Entropy formula for the asymptotic subrank of Dicke tensors
Let $n \ge 2$, let $d_1, \dots, d_n$ be positive integers and $k = d_1 + \cdots + d_n$. The Dicke tensor $$D_{\mathbf d} = \sum_{\substack{(i_1, \dots, i_k) \in \{1, \dots, n\}^k \\ |\{j : i_j = a\}| = d_a \text{ for all } a}} e_{i_1} \otimes \cdots \otimes e_{i_k} \in (\mathbb{C}^n)^{\otimes k}$$ corresponds to the monomial $x_1^{d_1} \cdots x_n^{d_n}$. Let $\widetilde Q(T) = \lim_{m \to \infty} Q(T^{\boxtimes m})^{1/m}$ be the asymptotic subrank of a tensor $T$, where $Q$ is subrank over $\mathbb{C}$ and $T^{\boxtimes m}$ is the $m$-th Kronecker power, and let $H(p_1, \dots, p_n) = -\sum_{a=1}^n p_a \log_2 p_a$. Vrana and Christandl showed that $\widetilde Q(D_{\mathbf d}) \le 2^{H(d_1/k, \dots, d_n/k)}$ for every $\mathbf d$, with equality for $\mathbf d = (k - 1, 1)$, the W-states (a gap in the proof was closed by Christandl, Vrana and Zuiddam). Equality also holds for $k = 3$, by Strassen's theorem on tight $3$-tensors, for $\mathbf d = (2, 2)$ (Christandl, Vrana and Zuiddam), and for $\mathbf d = (k/2, k/2)$ with $k$ even and large enough (Arunachalam, Vrana and Zuiddam).
Problem. Is $\widetilde Q(D_{\mathbf d}) = 2^{H(d_1/k, \dots, d_n/k)}$ for every $n \ge 2$ and all positive integers $d_1, \dots, d_n$?
Source: Vrana and Christandl (2015), Section VI, as a rate of conversion from Dicke to GHZ states; as a subrank formula, Christandl, Vrana and Zuiddam (2019), Question 1.3.3, and Oneto and Ventura (2025), Conjecture 4.26.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- CO — Combinatorics
- Secondary
- AG, NA
- Keywords
- asymptotic subranks, Dicke tensors, OpenProblemsInNLA
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{vrana2015,
author = {Vrana, P{\'e}ter and Christandl, Matthias},
title = {Asymptotic entanglement transformation between {W} and {GHZ} states},
journal = {J. Math. Phys.},
volume = {56},
number = {2},
year = {2015},
pages = {022204},
doi = {10.1063/1.4908106},
eprint = {1310.3244},
archivePrefix = {arXiv},
}
@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{strassen1991,
author = {Strassen, Volker},
title = {Degeneration and complexity of bilinear maps: some asymptotic spectra},
journal = {J. Reine Angew. Math.},
volume = {413},
year = {1991},
pages = {127--180},
doi = {10.1515/crll.1991.413.127},
}
@article{christandl2019,
author = {Christandl, Matthias and Vrana, P{\'e}ter and Zuiddam, Jeroen},
title = {Asymptotic tensor rank of graph tensors: beyond matrix multiplication},
journal = {Comput. Complexity},
volume = {28},
number = {1},
year = {2019},
pages = {57--111},
doi = {10.1007/s00037-018-0172-8},
eprint = {1609.07476},
archivePrefix = {arXiv},
}
@article{arunachalam2020,
author = {Arunachalam, Srinivasan and Vrana, P{\'e}ter and Zuiddam, Jeroen},
title = {The asymptotic induced matching number of hypergraphs: balanced binary strings},
journal = {Electron. J. Combin.},
volume = {27},
number = {3},
year = {2020},
pages = {P3.12},
doi = {10.37236/9019},
eprint = {1905.03148},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr28,
author = {Townsend, Alex},
title = {{TR-28}: Entropy formula for the asymptotic subrank of Dicke tensors},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-28/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00136, "Entropy formula for the asymptotic subrank of Dicke tensors", clopen.solutions.