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

TL;DR

This paper explores a quantum extension of Euler's classical problem, demonstrating how quantum superpositions and entanglement enable solutions in cases where classical combinatorial structures do not exist, through a geometric perspective.

Contribution

It introduces a geometric framework for understanding quantum Graeco-Latin squares, linking entanglement properties to non-displaceable manifolds and their intersections in complex projective space.

Findings

Existence of a quantum Graeco-Latin square of size six implies intersecting maximally entangled states.

Quantum approach allows solutions to classical impossibility results.

Geometric interpretation connects entanglement with non-displaceable manifolds.

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.