Thirty-six entangled officers of Euler: Quantum solution to a classically impossible

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

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FROM: https://www.livescience.com/math-puz...antum-solution Centuries-old 'impossible' math problem cracked using the strange physics of Schrödinger's cat By Stephanie Pappas (2022-01-20) A math problem developed 243 years ago can be solved only by using quantum entanglement, new research finds. The mathematics problem is a bit like Sudoku on steroids. It's called Euler's officer problem, after Leonhard Euler, the mathematician who first proposed it in 1779. Here's the puzzle: You're commanding an army with six regiments. Each regiment contains six different officers of six different ranks. Can you arrange them in a 6-by-6 square without repeating a rank or regiment in any given row or column? Euler couldn't find such an arrangement, and later computations proved that there was no solution. In fact, a paper published in 1960 in the Canadian Journal of Mathematics used the newfound power of computers to show that 6 was the one number over 2 where no such arrangement existed. Now, though, researchers have found a new solution to Euler's problem. As Quanta Magazine's Daniel Garisto reported, a new study posted to the preprint database arXiv finds that you can arrange six regiments of six officers of six different ranks in a grid without repeating any rank or regiment more than once in any row or column... if the officers are in a state of quantum entanglement. The paper, which has been submitted for peer review at the journal Physical Review Letters, takes advantage of the fact that quantum objects can be in multiple possible states until they are measured. (Quantum entanglement was famously demonstrated by the Schrödinger's cat thought experiment, in which a cat is trapped in a box with radioactive poison; the cat is both dead and alive until you open the box.) (SNIP)

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Original Paper:

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FROM: https://arxiv.org/abs/2104.05122 Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem Suhail Ahmad Rather, Adam Burchardt, Wojciech Bruzda, Grzegorz Rajchel-Mieldzioć, Arul Lakshminarayan, Karol Życzkowski - Submitted on 11 Apr 2021 (v1), last revised 6 Aug 2021 (this version, v2) The 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. As a consequence, we find an example of the long-elusive Absolutely Maximally Entangled state AME(4,6) of four subsystems with six levels each, equivalently a 2-unitary matrix of size 36, which maximizes the entangling power among all bipartite unitary gates of this dimension, or a perfect tensor with four indices, each running from one to six. This special state deserves the appellation golden AME state as the golden ratio appears prominently in its elements. This result allows us to construct a pure nonadditive quhex quantum error detection code ((3,6,2))6, which saturates the Singleton bound and allows one to encode a 6-level state into a triplet of such states... (SNIP)

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