This article has been reviewed according to Science X's editorial process and policies. Editors have highlighted the following attributes while ensuring the content's credibility: Latin squares are arrangements of symbols in a grid in which every symbol appears exactly once in each row and column. These symbol arrangements, which were first studied more than three centuries ago, are now widely used to optimize experimental designs and develop secure cryptographic systems, puzzles or other complex combinatorial structures.
In 1782, the Swiss mathematician Leonhard Euler devised a renowned mathematical problem based on Latin squares, known as the 36 officers problem. This problem entails arranging 36 officers from six regiments and six ranks in a 6-by-6 square grid, ensuring that every row and column contains one officer from each regiment and each rank. In the centuries after Euler introduced this problem, mathematicians showed that it could not be solved using classical approaches.
More recently, theorists introduced quantum versions of this problem, replacing the individual symbols in ordinary Latin squares with mathematical descriptions of possible quantum system states. Researchers at Polytechnic University of Catalunya set out to investigate whether quantum Latin squares could be used to solve the 36 officers problem without relying on entanglement (i.e., a quantum phenomenon linking two or more particles in such a way that measuring the state of one instantly dictates the states of the others). Their paper, published in Physical Review Letters, shows that entanglement is essential for solving a quantum version of the problem.
"We wanted to understand better the recent quantum solution to Euler's 36 officers problem, proposed in a paper by Rather et al." Robin Simoens, corresponding author of the paper, told Phys.org. "Curiously, for that solution to work, the ranks of officers must assume multiple values at the same time, depending on each other. This dependence is called entanglement.
We wanted to know whether there exists a simpler 'in-between' solution that does not require the complication of entanglement. This can be compared to a Sudoku where you are allowed to write multiple numbers in one square." From a practical standpoint, a quantum solution to Euler's problem that does not rely on entanglement would enable the generation of a given state using circuits with a lower gate depth. Yet Simoens and his colleague Simeon Ball showed mathematically that such a solution does not exist.
The researchers considered two Latin squares with six rows and six columns. Each square contained six symbols, and every symbol appeared once in every row and once in every column. A further condition they set was that the two Latin squares should be orthogonal, meaning that superimposing them would produce every possible ordered pair of symbols exactly once.
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