Thirty-six entangled officers of Euler: quantum solution of a classically impossible problem




Center for Theoretical Physics and Jagiellonian University

April 27, 2022 12:30 PM

This will be a hybrid colloquium:

onsite meeting: Audytorium of IFPAN, Al. Lotnikow 32/46

online version:


Classical combinatorial designs are composed of elements of a finite set and arranged with a certain symmetry and balance. A simple example of a combinatorial design is given by a single Latin square: square array of size d filled with d copies of d different symbols, each occurring once in each row and in each column. Such patterns are useful in statistics to design optimal experiments.

Analogous collections of quantum states, called  a quantum design, determine distinguished quantum measurements and can be applied for various purposes of 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.

As a consequence, we find  an example of Absolutely Maximally Entangled (AME) state of four subsystems with six levels each, which deserves the appellation golden AME state, as the golden ratio appears prominently in its elements. This state enables us to construct a pure nonadditive quhex quantum error detection code, which allows one to encode a 6-level state into a triplet of such states. Furthermore, using such a state one can teleport any unknown, two-dice quantum state, from any two owners of two subsystems to the lab possessing the two other dice forming the four-dice system.


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

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

[3] Ph. Ball,  A Quantum Solution to an 18th-Century Puzzle, Physics, 15, 29 (2022);

[4] K. Życzkowski, W. Bruzda, G. Rajchel-Mieldzioć, A.Burchardt, S.A Rather, A. Lakshminarayan, 9 × 4 = 6 × 6: Understanding the quantum solution to the Euler's problem of 36 officers, preprint arXiv:2204.06800