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Centuries-old 'impossible' math problem cracked using the bizarre physics of Schrödinger's cat

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A 243-year-old arithmetic issue can only be solved through quantum entanglement, according to a new study.

The arithmetic puzzle is similar to Sudoku on steroids. The problem is known as Euler's officer problem after Leonhard Euler, a mathematician who introduced it in 1779. Here's the riddle: You lead a force of six regiments. Each regiment has six distinct officers of various ranks. Is it possible to arrange them in a 6-by-6 square without repeating a rank or regiment in any given row or column?

Euler was unable to find such an arrangement, and subsequent computations demonstrated that there was no solution. A publication published in the Canadian Journal of Mathematics in 1960 used the newly discovered capability of computers to demonstrate that 6 was the only number greater than 2 where no such arrangement existed.

However, experts have discovered a novel solution to Euler's conundrum. According to Quanta Magazine's Daniel Garisto, a new study published on the preprint database arXiv demonstrates 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 research, which has been submitted to the journal Physical Review Letters for peer review, takes advantage of the fact that quantum objects can exist in numerous states until they are measured. (The Schrödinger's cat thought experiment notably proved quantum entanglement, in which a cat is locked in a box containing radioactive poison; the cat is both dead and alive until the box is opened.)

Each officer in Euler's classic dilemma has a fixed regiment and rank. They could be a first lieutenant in the Red Regiment or a captain in the Blue Regiment, for example. (Colors are occasionally used to help identify regimens when visualising the grids.)

A quantum officer, on the other hand, may serve in more than one regiment or rank at the same time. A single officer might be a first lieutenant in the Red Regiment, a captain in the Blue Regiment, a major in the Green Regiment, or a colonel in the Purple Regiment. (Or, in theory, any other combination.)

The fact that the policemen on the grid can be in a state of quantum entanglement is the key to solving Euler's issue with this identity switcheroo. The state of one object informs the state of another in entanglement. If Officer No. 1 is a first lieutenant in the Red Regiment, Officer No. 2 must be a major in the Green Regiment, and vice versa.

The authors of the new work, led by Adam Burchardt, a postdoctoral researcher at Jagiellonian University in Poland, demonstrated using brute-force computer power that saturating the grid with quantum cops enabled the answer. Surprisingly, the entanglement has its pattern, according to research co-author Suhail Rather, a physicist at the Indian Institute of Technology Madras. Officers are only entangled with officers one step lower or higher in rank, and regiments are only entangled with nearby regiments.

According to Quanta Magazine, the findings could have a real influence on quantum data storage. Entangled states can be employed in quantum computing to ensure that data remains safe even if a mistake occurs, a technique known as quantum error correction. 

The researchers discovered a maximally entangled state by entangling 36 quantum cops in a state of interdependent interactions. Such states may be critical for quantum computing's durable data storage.