Absolutely Maximally Entangled States of 2q Parties in Every Odd Prime-Power Dimension q
Abstract
Absolutely maximally entangled (AME) states represent an extreme form of multipartite entanglement: every reduced system containing at most half of the parties is maximally mixed. These states provide perfect tensors and optimal quantum error-correcting codes, yet their existence is known only in restricted parameter regimes. For every odd prime power q=pe≥3, we construct a stabilizer AME(2q,q) state whose normalized one-party projection yields a stabilizer AME(2q−1,q) state. A closed-form q× q bordered-circulant matrix Aq over 𝔽q2 generates a Hermitian self-dual maximum distance separable (MDS) code [2q,q,q+1]q2, which lies outside the extended Reed-Solomon classes. In suitable bases, the amplitude tensors define normalized q-unitary complex Hadamard matrices of order qq with pth-root phases. Additional constructions yield AME(q+3,q) states and families at intermediate particle numbers through explicit rescalings of selected submatrices. We also provide nine explicit parent matrices and the corresponding one-party projections.
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BibTeX
@misc{hontarenko2026absolutely,
title = {{Absolutely Maximally Entangled States of $2q$ Parties in Every Odd Prime-Power Dimension $q$}},
author = {Mykhailo Hontarenko and Karol Życzkowski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11796},
eprint = {2609.11796},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11796},
}
RIS
TY - UNPB
TI - Absolutely Maximally Entangled States of 2q Parties in Every Odd Prime-Power Dimension q
AU - Mykhailo Hontarenko
AU - Karol Życzkowski
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11796
UR - https://arxiv.org/abs/2609.11796
AB - Absolutely maximally entangled (AME) states represent an extreme form of multipartite entanglement: every reduced system containing at most half of the parties is maximally mixed. These states provide perfect tensors and optimal quantum error-correcting codes, yet their existence is known only in restricted parameter regimes. For every odd prime power q=p^e≥3, we construct a stabilizer AME(2q,q) state whose normalized one-party projection yields a stabilizer AME(2q−1,q) state. A closed-form q× q bordered-circulant matrix A_q over 𝔽_q^2 generates a Hermitian self-dual maximum distance separable (MDS) code [2q,q,q+1]_q^2, which lies outside the extended Reed-Solomon classes. In suitable bases, the amplitude tensors define normalized q-unitary complex Hadamard matrices of order q^q with pth-root phases. Additional constructions yield AME(q+3,q) states and families at intermediate particle numbers through explicit rescalings of selected submatrices. We also provide nine explicit parent matrices and the corresponding one-party projections.
ER -