Euler's thirty-six officers need not all be entangled: real AME(4,6), families and classification
Abstract
Absolutely maximally entangled states of four six-level systems, AME(4,6), are the quantum solutions of Euler's problem of the thirty-six officers. Until recently all reported solutions were complex, and it was conjectured that no real orthogonal 2-unitary matrix of order $36$ exists. We give real AME(4,6) states with $60$ non-zero coefficients in {±1,√2}/√(72). Of their thirty-six officers, $12$ are classical, as some must be whenever fewer than $72$ coefficients are non-zero. They lie in the −1 eigenspace of an order-$4$ local symmetry, and on their support the AME conditions become linear. We derive two closed-form families of AME(4,6) states that meet at one of these states: a $10$-parameter family that is supported on $72$ basis states, and an $11$-parameter family on $60$ basis states. At each of its two ends the first family loses $12$ coefficients, and the AME states on the remaining $60$ basis states form an $11$-parameter family: the second family at one end, and an analogous family at the other. These two $60$-term families contain $4096$ real AME states up to diagonal local unitaries, $1024$ of which are limits of the first family. They fall into exactly $20$ classes under local unitaries and permutations of the parties, and we certify their separation with interval arithmetic.
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BibTeX
@misc{pandya2026eulers,
title = {{Euler's thirty-six officers need not all be entangled: real AME(4,6), families and classification}},
author = {Palash Pandya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16673},
eprint = {2610.16673},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16673},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euler's thirty-six officers need not all be entangled: real AME(4,6), families and classification
AU - Palash Pandya
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16673
UR - https://arxiv.org/abs/2610.16673
AB - Absolutely maximally entangled states of four six-level systems, AME(4,6), are the quantum solutions of Euler's problem of the thirty-six officers. Until recently all reported solutions were complex, and it was conjectured that no real orthogonal 2-unitary matrix of order $36$ exists. We give real AME(4,6) states with $60$ non-zero coefficients in {±1,√2}/√(72). Of their thirty-six officers, $12$ are classical, as some must be whenever fewer than $72$ coefficients are non-zero. They lie in the −1 eigenspace of an order-$4$ local symmetry, and on their support the AME conditions become linear. We derive two closed-form families of AME(4,6) states that meet at one of these states: a $10$-parameter family that is supported on $72$ basis states, and an $11$-parameter family on $60$ basis states. At each of its two ends the first family loses $12$ coefficients, and the AME states on the remaining $60$ basis states form an $11$-parameter family: the second family at one end, and an analogous family at the other. These two $60$-term families contain $4096$ real AME states up to diagonal local unitaries, $1024$ of which are limits of the first family. They fall into exactly $20$ classes under local unitaries and permutations of the parties, and we certify their separation with interval arithmetic.
ER -