More than 240 years ago, the famous mathematician Leonhard Euler posed a question: if six army regiments each had six officers of six different ranks, could they arranged in a square formation without rows or columns repeating a rank or regime?
After searching for a futile solution, Euler declared the problem impossible and more than a century later, the French mathematician Gaston Tarry proved him right. Then, 60 years passed thatwhen the advent of computers removed the need for painstakingly testing every possible combination by hand, mathematicians Parker, Bose, and Shrikhande proved an even stronger result: not only the six-by-six square is impossible, but the only size of the square except two-by-two which has no solution at all.
Now, in mathematics, once a theorem is proven, it is proven forever. So it may come as a surprise to learn that a 2022 paper, published in the journal Physical Review Letters, apparently found a solution. There’s just one catch: the officers must be in a state of quantum entanglement.
I think their paper is wonderful, quantum physicist Gemma De las Cuevas, who was not involved in the work, told Quanta Magazine at the time. There’s a lot of quantum magic out there. And not only that, but you will feel the whole role [the authors] love is a problem.
To explain what’s going on, let’s start with a classical example. The Eulers 36 Officers problem, as it is known, is a special type of magic square called an orthogonal Latin square think of it like two sudokus that you have to solve at the same time on the same grid. For example, a four-by-four orthogonal Latin square would look like this:
No color repeats in any direction; no number repeats in any direction; all numbers of all colors are represented.
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In each square of the grid defined like this with a specific number and a specific color Eulers original six-by-six problem is impossible. However, in the quantum world, things are more flexible: things are in superpositions in the states.
In basic terms or at least, as basic as it gets when talking about quantum physics this means that any given general can rank in many regiments at the same time. Using our colorful double-sudoku example, we can imagine a grid square filled with, say, a superposition of a green two and a red one.
Reen what? Grade Tone?
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Now, researchers think, Eulers problem has a solution. But what was it?
At first glance, it seems like the team has their work cut out for them. Not only do they have to solve a six-by-six double sudoku that is known to be impossible in the classical setting, but now they have to do it in multiple dimensions at once.
Luckily, however, they had two things on their side: first, a classical near solution that they could use as a jumping-off point, and second, the seemingly mysterious property of quantum entanglement.
Simply put, two states are said to be linked if one state tells you about the other. As a classical analogy, imagine that you know that your friend has two children, A and B (your friend is not good with names) of the same sex. That means that knowing that baby A is a girl tells you for sure that baby B is also a girl so that both genders of the children are related.
Entanglement does not always work well, where one state tells you absolutely everything about the other but when it does, it is called an absolutely maximally entangled (AME) state. Another example might be flipping coins: if Alice and Bob each flip a coin and Alice gets heads, then if the coins are heads, Bob knows that he didn’t get heads. -well he has tails, and vice versa.
Remarkably, the solution to this quantum officer problem turns out to have this property and this is where it gets interesting. See, the above example works for two coins, and for three, but for four, it is impossible. But the problem with 36 Officers isn’t like flipping dice, the authors realize it’s like rolling entangled dice.
[Imagine that] Alice chooses any two dice and rolls them, getting one of the 36 equally possible outcomes, while Bob rolls the rest. If the entire state is AME, Alice can always figure out the result that Bobs gets as part of the four-party system, the paper explains.
In addition, such a state allows one to teleport to any unknown, two-dice quantum state, from any two owners of two subsystems to the lab with two other dice in the linked state of the four-party system, the authors continued. This is not possible if the dice are replaced by two-sided coins.
Since these AME systems are often explained using orthogonal Latin squares, researchers already know that they exist for four people throwing dice with any number of sides. anything, that is, except two or six. Remember: those orthogonal Latin squares do not exist, so they cannot be used to prove that there is an AME state in that dimension.
However, by finding a solution to Eulers 243-year-old problem, the researchers did something amazing: they found a four-party AME system with six dimensions. In doing so, they may even have discovered a new type of AME that has no analog in a classical system.
Euler claimed in 1779 that no solution existed. The proof, of Tarry, came only 121 years later in 1900, the authors wrote. After another 121 years, we present a quantum version solution that officials can relate to.
The quantum design presented here is likely to trigger further research in the new field of quantum combinatorics, they conclude.
The study was published in Physical Review Letters.
An earlier version of this article was published in January 2022.
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