Fractional and quantum Latin squares: real solutions of quantum 36 officers problem
Abstract
A Latin square can be extended by replacing its symbols with probabilities or quantum states. We relate these two extensions through explicit examples, from classical and fractional Latin squares to quantum Latin squares and a real solution of Euler's quantum 36 officers problem. A simultaneous cyclic-shift ansatz on a four-torus of tensor coefficients gives a real absolutely maximally entangled (AME) state of four six-level systems. A local orthogonal change of basis yields a representative with 96 nonzero amplitudes. Two single-party arrays obtained from the corresponding quantum orthogonal Latin square yield fractional Latin squares. Signs matter: comparison with the independent real construction of Bruzda and its sign variant distinguishes three local-unitary classes, two of which have identical probabilities. We extend this probability representation to existing seven- and nine-party AME states with six levels each. Their amplitude matrices reveal 48 complex Hadamard blocks of order 33 and six of order 64, respectively. These constructions separate what can be read from a probability picture from the coherent conditions responsible for maximal entanglement.
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BibTeX
@misc{hontarenko2026fractional,
title = {{Fractional and quantum Latin squares: real solutions of quantum 36 officers problem}},
author = {Mykhailo Hontarenko and Karol Życzkowski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17170},
eprint = {2610.17170},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17170},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fractional and quantum Latin squares: real solutions of quantum 36 officers problem
AU - Mykhailo Hontarenko
AU - Karol Życzkowski
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17170
UR - https://arxiv.org/abs/2610.17170
AB - A Latin square can be extended by replacing its symbols with probabilities or quantum states. We relate these two extensions through explicit examples, from classical and fractional Latin squares to quantum Latin squares and a real solution of Euler's quantum 36 officers problem. A simultaneous cyclic-shift ansatz on a four-torus of tensor coefficients gives a real absolutely maximally entangled (AME) state of four six-level systems. A local orthogonal change of basis yields a representative with 96 nonzero amplitudes. Two single-party arrays obtained from the corresponding quantum orthogonal Latin square yield fractional Latin squares. Signs matter: comparison with the independent real construction of Bruzda and its sign variant distinguishes three local-unitary classes, two of which have identical probabilities. We extend this probability representation to existing seven- and nine-party AME states with six levels each. Their amplitude matrices reveal 48 complex Hadamard blocks of order 3^3 and six of order 6^4, respectively. These constructions separate what can be read from a probability picture from the coherent conditions responsible for maximal entanglement.
ER -