Euler's thirty-six officers need not all be entangled: real AME(4,6), families and classification

In the quantum world, particles can become linked in ways that defy our everyday experience, a phenomenon known as entanglement. When a group of particles is perfectly entangled, it means that no matter how you split the group into two smaller parts, each part looks completely random and uncorrelated on its own, yet they remain perfectly coordinated as a whole. Scientists call these special configurations "absolutely maximally entangled" states. They are not just mathematical curiosities; they serve as the foundation for secure quantum communication and are essential for building the next generation of quantum computers. For decades, researchers have been hunting for a specific type of these states involving four groups of six options each. This search is deeply connected to a famous puzzle from the 18th century posed by the mathematician Leonhard Euler, who asked if it was possible to arrange thirty-six officers of six different ranks and six different regiments in a square grid so that no rank or regiment is repeated in any row or column. Euler believed it was impossible, and for a long time, it was thought that no quantum solution could exist either, especially one made of simple, real numbers rather than complex, abstract ones.

For a long time, the only known solutions to this quantum puzzle were complex, meaning they relied on numbers that include the square root of negative one, a concept that has no direct counterpart in the physical world of real numbers. This led many experts to suspect that a solution using only real numbers might be impossible. However, a new study has definitively overturned that suspicion. Researchers have now constructed a specific quantum state that solves Euler's puzzle using only real numbers. This state involves four parties, each with six levels, and it is the first time such a solution has been found that does not rely on complex coefficients. The discovery confirms that real solutions exist exactly when complex ones do, bridging a gap that had stood for years.

The author behind this discovery, Palash Pandya, found a solution that is remarkably sparse, meaning it uses fewer building blocks than any previously known version. While earlier solutions required a large number of non-zero components, this new state needs only sixty. To visualize this, imagine the thirty-six officers arranged in a grid. In this new solution, twelve of the officers are "classical," meaning they exist in a single, definite state, while the remaining twenty-four are "entangled," existing as a blend of two possibilities at once. This mix is necessary because the state is so sparse; if there were fewer than seventy-two components, some officers would have to be classical. The author showed that these twelve classical officers are not random; they sit in a specific mathematical space defined by a local symmetry, which allowed the author to simplify the problem from a complex, non-linear challenge into a set of linear equations that could be solved directly.

Using this insight, the author did not just find one solution; they discovered entire families of solutions. They described two continuous families of states that meet at the specific real solution they found. One family is supported on seventy-two basis states, while the other is supported on the sixty states of the sparse solution. These families act like a landscape of possibilities, where the author can move smoothly from one type of state to another. At the ends of these families, they found that the states become real and sparse again, but with a different arrangement of the classical officers. By exploring these families, the author was able to classify every possible real solution of this type. They found that there are exactly four thousand and ninety-six distinct real states, which fall into exactly twenty different classes based on how they can be transformed into one another through local changes and permutations.

The author verified their findings with extreme precision, using computer scripts to check every mathematical detail. They confirmed that these states are isolated points, meaning you cannot smoothly deform them into other real states without breaking the rules of entanglement. This isolation is significant because it suggests these states are unique and robust. The study also compared their findings with other recently discovered real solutions, showing that while some are equivalent to their work, others belong to entirely different categories with different properties. For instance, one other known real solution has ninety-six components and behaves differently, lacking the classical officers found in this new work. This distinction highlights that the landscape of quantum solutions is rich and varied, with different types of states serving different potential roles in quantum technology.

The implications of this work extend beyond solving an old puzzle. These real, maximally entangled states are perfect for building "dual-unitary" gates, which are powerful tools for creating quantum circuits that can process information efficiently. Because these states use real numbers, they are potentially easier to implement in physical systems where complex numbers are harder to realize directly. The discovery proves that the quantum world can accommodate these perfect, real-numbered structures, opening new doors for designing quantum codes and holographic models. The author has made their data and the code used to verify their results publicly available, allowing others to build upon this foundation. By turning a centuries-old combinatorial puzzle into a concrete quantum reality, this work not only answers a long-standing question but also provides a new toolkit for the future of quantum engineering.