Quantum version of the Euler's problem: a geometric perspective

• URL: http://arxiv.org/abs/2212.03903v2
• Date: Fri, 23 Dec 2022 15:01:28 GMT
• Title: Quantum version of the Euler's problem: a geometric perspective
• Authors: Karol Zyczkowski
• Abstract summary: We analyze the recently found solution to the quantum version of the Euler's problem from a geometric point of view. Existence of a quantum Graeco-Latin square of size six, equivalent to a maximally entangled state of four subsystems with d=6 levels each, implies that three copies of the manifold U(36)/U(1) of maximally entangled states of the $36times 36$ system, embedded in the complex projective space $CP36times 36 -1$, do intersect simultaneously at a certain point.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The classical combinatorial problem of $36$ officers has no solution, as there are no Graeco-Latin squares of order six. The situation changes if one works in a quantum setup and allows for superpositions of classical objects and admits entangled states. We analyze the recently found solution to the quantum version of the Euler's problem from a geometric point of view. The notion of a non-displaceable manifold embedded in a larger space is recalled. This property implies that any two copies of such a manifold, like two great circles on a sphere, do intersect. Existence of a quantum Graeco-Latin square of size six, equivalent to a maximally entangled state of four subsystems with d=6 levels each, implies that three copies of the manifold U(36)/U(1) of maximally entangled states of the $36\times 36$ system, embedded in the complex projective space ${C}P^{36\times 36 -1}$, do intersect simultaneously at a certain point.

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