More than 240 years ago, the famous mathematician Leonhard Euler posed a question. Suppose there are six officers of six different ranks in each of the six military regiments. Shouldn't they also be arranged in a square and no row or column should overlap a rank and its own regiment? In fact, it's similar to the combination questions we often ask our students at the high school level. There is no doubt that it is one of the most complicated subjects in mathematics. Even seasoned math teachers can't help but wonder if there are combination questions. In our article, it is about exactly this, “Quantum Entanglement Solves the 243-Year-Old Puzzle”. Let's start.

Leonhard Euler, after vainly searching for a solution, declared the problem impossible. More than a century later, French mathematician Gaston Tarry proved him right. In other words, we wanted to share the above image with you in order to be a little more understandable.

Sixty years from now, when the advent of computers eliminated the need to laboriously test every possible combination by hand, mathematicians Parker, Bose, and Shrikhande proved an even stronger result.

Not only is six-by-six square impossible, it's the only square size that has no other than two-by-two solution.

In mathematics, once a theorem is proven, it is thought to last forever.

So it's probably already surprising to learn that a new paper, now available as a preprint and submitted to the journal Physical Review Letters, has apparently found a solution.

On the other hand, there is a trick of the event. The officers involved in the combination have to exist in a state of quantum entanglement.

Quantum physicist Gemma De las Cuevas told Quanta Magazine:

“I think the article they did is very nice,” he said.

“There's a lot of quantum magic out there. Not only that, you can deeply feel how they approached the current problem as you read the article.”

Let's start with a classic example to explain what's going on. Euler's "36 Officers" problem, as is known, is a special type of magic square called the "orthogonal Latin square".

Think of it like two sudoku you have to solve at the same time on the same grid. For example, a four-by-four orthogonal Latin square might look like this.

With every square in the grid defined in this way – with a fixed number and a fixed color – Euler's original six-by-six problem is impossible. However, things are more flexible in the quantum world. Each state is found in superpositions of states. So we call it superposition in physics.

In basic terms it means that any general can hold more than one rank in more than one regiment at the same time.

Using our color double sudoku example, we can imagine that a square in the grid is filled with a superposition of green two and a red one.

Now, the researchers thought there would be a solution to Euler's problem. But what was it?

At first glance, it may seem that the team is making their job much more difficult. They had to solve a six-by-six double sudoku, which was known to be impossible in the classical setting, but they also had to do it in multiple dimensions simultaneously.

Fortunately, they had a few things with them.

First, a classical close solution that they can use as a jumping off point, and second, the seemingly mysterious property of quantum entanglement.

Simply put, the two situations are said to be confused when one situation tells you something about the other.

As a classic analogy, imagine you know that your friend has two children of the same gender, A and B (who is not very good at remembering names).

This means that knowing child A is a girl tells you for sure that child B is also a girl. In other words, the genders of the two children are now mixed.

Entanglement doesn't always turn out so well, one situation tells you absolutely everything about the other. But when it does, it's definitely called a maximum entanglement (AME) state.

Another example would be tossing a coin. If Alice and Bob flip a coin and Alice sees that she is holding heads, if the coins are entangled, Bob will know that it is a tail without looking, and vice versa.

The above example works for both coins and for three but that's impossible for four. But the article's authors realized that the 36 Officers problem is not like rolling dice. It was more like rolling entangled dice.

Now if we imagine Alice picking any two dice and rolling them, Bob gets one of 36 outcomes with the same probability as he rolls the rest. If the whole case is [AME], Alice can always derive the result obtained on Bob's part of the 4-party system,” the article explains.

“Furthermore, such a state allows any unknown, two-membrane quantum state to be teleported from any two owners of the two subsystems to the laboratory with the other two membranes of the entangled state of the four-party system,” the authors continue.

“These are not possible if the dice are replaced by two-sided coins.”

Because these AME systems can often be explained using orthogonal Latin squares, the researchers already knew that they exist for any number of people, namely four who roll any number of dice other than two or six.

Remember: these upright Latin squares do not exist, so they cannot be used to prove the existence of an AME condition of that size.

However, by finding a solution to Euler's 243-year-old problem, the researchers did something wonderful: they found an AME system consisting of six-dimensional four-partitions. In doing so, they may even have discovered a whole new type of AME that has no analogue in a classical system.

“Euler … claimed in 1779 that there was no solution. The first paper with proof of this statement came by Tarry only 121 years later, in 1900,” the authors write. “After 121 years, we have presented a solution to the entangled quantum version of the officers.”

“It is tempting to believe that the quantum design presented here will trigger further research on quantum combinatorics,” they conclude.

Source: IFL

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