XXXIX Workshop on Geometric Methods in Physics 19.06–25.06.2022 XI School on Geometry and Physics 27.06–1.07.2022
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In order to secure the right conditions for our Workshop we have decided to exceptionally move (only for this year's meeting) the site of our Workshop to the campus of our University in Białystok.

# Karol Zyczkowski

## Extremal quantum states & combinatorial designs

A quantum combinatorial design is composed of quantum states, arranged with a certain symmetry and balance. Such a constellation> of states determines distinguished quantum measurements and can be applied for quantum information processing. Negative solution to the famous problem of $36$ officers of Euler implies that there are no two orthogonal Latin squares of order six. We show that the problem has a solution, provided the officers are entangled, and construct orthogonal quantum Latin squares of this size. The solution can be visualized on a chessboard of size six, which shows that $36$ officers are split in nine groups, each containing of four entangled states. It allows us to construct a pure nonadditive quhex quantum error detection code and four-party states with extremal entanglement properties.

References:
[1] S.A Rather, A.Burchardt, W. Bruzda, G. Rajchel-Mieldzioć, A. Lakshminarayan and K. Życzkowski, Thirty-six entangled officers of Euler,
{\sl Phys. Rev. Lett.} {\bf 128}, 080507 (2022).

[2] D. Garisto, Euler’s 243-Year-Old ‘Impossible’ Puzzle Gets a Quantum Solution, Quanta Magazine, Jan. 10, 2022; https://www.quantamagazine.org/

 Event sponsored by: University of Bialystok

Webpage by: Tomasz Golinski