In 1779, the Swiss mathematician Leonhard Euler posed a puzzle that has since become famous. Can the 36 officers be arranged in a square so that no row or column repeats a rank?
The puzzle can be solved if there are five ranks and five units. After searching for a solution for the case of 36 officers, Euler concluded that it was impossible. The French mathematician, Gaston Tarry, proved that there was no way to arrange the 36 officers in a square without repetition. In 1960, mathematicians used computers to prove that solutions exist for any number of units and ranks greater than two.
People have been entranced by similar puzzles for more than 2,000 years. Cultures around the world have made magic squares, which are an array of numbers that add to the same sum along each row and column, and Latin squares, which are filled with symbols that appear once per row and column. The squares have been used in art and urban planning. One popular Latin square has sub squares that don't have repeating symbols. The Latin square has rules for ranks and regiments, and the 36 officers puzzle asks for an "orthogonal Latin square" in which both of those properties satisfy the rules.
The game has changed since Euler thought there was no such square. A group of quantum physicists in India and Poland submitted a paper to the Physical Review Letters that demonstrates that it is possible to arrange 36 officers in a way that complies with the criteria set by Euler. The quantum versions of magic square and Latin square puzzles are fun and games, but have applications for quantum communication and quantum computing.
The University of Innsbruck's quantum physicist, who was not involved with the work, said the paper was beautiful. There is a lot of magic in there. You can feel their love for the problem in the paper.
The new era of quantum puzzling began in 2016 when Jamie Vicary of the University of Cambridge and his then-student Ben Musto had the idea that the entries in Latin squares could be made quantum.
There are promising ideas in science and mathematics. Join the conversation with us.
In quantum mechanics, objects can be in a position of multiple states, such as here and there, or both up and down. At which point they settle on one state, quantum objects stay in this limbo until they are measured. Latin squares are quantum states that can be in superpositions. A quantum state is represented by a long and straight arrow. A superposition is an arrow that is formed by combining multiple variables. The quantum states along each row or column of a Latin square must correspond to the same ones in the other side of the square.
Physicists and mathematicians were interested in the strange properties of quantum Latin squares. Ion Nechita and Jordi Pillet created a quantum version of a game called SudoQ. The rows, columns, and sub squares have nine parallel lines instead of the usual 0 through 9.
Adam Burchardt, a researcher at Jagiellonian University in Poland, and his colleagues reexamined the puzzle about the 36 officers. They wondered if the officers of Euler were made quantum.
Each entry is an officer with a well-defined rank. It is helpful to think of the 36 officers as colorful chess pieces, whose rank can be king, queen, rook, bishop, knight or pawn, and whose regiment is represented by red, orange, yellow, green, blue or purple. In the quantum version, officers are formed from the positions of ranks and armies. An officer could be a red king and an orange queen.
The officers are composed of quantum states that have a special relationship with each other. If the king is red and the queen is orange, you can tell immediately that the queen is orange. It is because of the peculiar nature of entanglement that officers along each line can be in a straight line.
To prove the theory worked, the authors had to build an array filled with quantum officers. They had to rely on the computer for help with a lot of configurations. The researchers plugged in a classical near-solution that had 36 classical officers with only a few repeats of ranks in a row or column, and applied an algorithm that made the arrangement more quantum. The solution to a Rubik's cube is similar to this one, where you fix the first row, then the first column, and so on. The puzzle array was closer to being a true solution when they repeated the algorithm over and over. The researchers were able to see the pattern and fill in the few remaining entries by hand.
In the 18th century, he couldn't have known about the possibility of quantum officers.
Nechita said that the book on the problem is already closed. I like the way they get it.
One surprising feature of their solution was that officer ranks are only entangled with adjacent ranks and not with each other. The entries of the quantum Latin square have coefficients. The coefficients are numbers that tell you how much weight to give a term. The golden ratio is the ratio of coefficients that the algorithm landed on.
The solution is also known as an Absolutely Nested state, an arrangement of quantum objects that is important for a number of applications including quantum error-correction, so that it survives even if there is a problem. In an AME, correlations between measurement of quantum objects are as strong as they can be, if Alice and Bob have entangled coins, and Alice tosses her coin and gets heads, she knows for sure that Bob has tails, and vice versa. If Carol and Dave join the coin toss, Alice can never be sure what Bob gets.
The new research shows that if you have a set of four dice, they can be maximally entangled. The six-sided dice are similar to the Latin square. The researchers dubbed this a "Golden AME" because of the golden ratio's presence in their solution.
De las Cuevas said it was highly nontrivial. They give the state an analysis of it, not only that it exists.
Classical error-correcting codes and quantum versions have previously been found to be analogous to AMEs. The golden AME has no classical cryptographic analogue. It could be the first of a new class of codes. If the golden AME remains unique, it might be equally interesting.
The author of this article is related to an editor at Physical Review Letters, where the quantum Latin squares paper was submitted for publication. The two did not discuss the paper.
All Rights Reserved © 2024
