Thirty-six Entangled Officers of Euler: Quantum Solution to a Classically Impossible Problem.

Area of Science:

• Quantum Information Science
• Quantum Computing
• Quantum Error Correction

Background:

• Euler's 36 officers problem, a classic combinatorial puzzle, was famously proven to have no solution concerning orthogonal Latin squares of order six.
• Orthogonal Latin squares are fundamental in combinatorics and have applications in experimental design and coding theory.
• Absolutely Maximally Entangled (AME) states are crucial resources in quantum information processing, enabling advanced protocols like quantum error correction and quantum communication.

Purpose of the Study:

• To demonstrate a solution to Euler's 36 officers problem by incorporating quantum entanglement.
• To construct orthogonal quantum Latin squares of order six.
• To discover a novel Absolutely Maximally Entangled state AME(4,6) and a perfect tensor with four indices.

Main Methods:

• Utilized quantum entanglement to construct orthogonal quantum Latin squares of order six, thereby providing a solution to Euler's problem.
• Derived an example of the AME(4,6) state, a 2-unitary matrix of size 36, which represents a perfect tensor.
• Investigated the properties of the discovered AME state, noting the prominent appearance of the golden ratio in its elements, leading to the 'golden AME state' appellation.

Main Results:

• Successfully constructed orthogonal quantum Latin squares of order six, offering a quantum solution to Euler's problem.
• Discovered and characterized the golden AME state (AME(4,6)), a maximally entangling bipartite unitary gate of dimension 36.
• Developed a pure nonadditive quhex quantum error detection code ((3,6,2))_6 that saturates the Singleton bound.

Conclusions:

• The introduction of quantum entanglement provides a novel approach to solving long-standing combinatorial problems like Euler's 36 officers problem.
• The golden AME state is a significant resource for quantum information processing, particularly for maximizing entangling power.
• The newly constructed quantum error detection code offers enhanced capabilities for encoding and protecting quantum information in six-level systems.