Jan 12, 2022

What is this 240-year-old problem all about?

Leonhard Euler (1707 - 1783), Swiss mathematician and physicist, is most popularly known for his glorious equation called Euler's Identity: *e ^{iφ}* + 1 = 0, depicted below.

^{Geometric interpretation of Euler's identity, where i represents the imaginary axis of the complex plane and φ is the angle.}

Euler’s contributions in mathematics have been indispensable for the development of physics, particularly in quantum mechanics. As if that were not enough, now the quantum solution to Euler’s puzzle will probably mark a milestone in quantum computation, and in information theory. The puzzle as such consists of the following: Euler had examined the problem of having six different regiments, each with six officers of different ranks, and he wondered if these 36 officers can be arranged in a 6x6 square, so that each row and column contains one officer of each rank and regiment, without repeating rank or regiment in any row or column ...

For example, in a hypothetical situation for dimension 4 (d = 4, i.e., 4 categories and 4 subcategories), where the categories were the shape of the object (square, circle, triangle, and star) and the subcategories were colors (blue, green, red, and yellow), the solution where neither color nor shape is repeated in any row or column, and where, in addition, each shape has a different color, is shown below:

^{Image by Inés Urdaneta, for Resonance Science Foundation}

However, this condition is not fulfilled for the diagonals, and interestingly, the two main diagonals of the 6x6 square, both repeat the shape (all 4 places are occupied by squares on one main diagonal, and stars in the remaining intersecting main diagonal). And, since both main diagonals consist of the same shape, colors do not repeat throughout the diagonal. We will say that this solution results in a "diagonalization".

Euler observed that such an arrangement was impossible for dimension d = 6. Much later, in 1960, and with the help of computers, mathematicians proved that solutions exist for any number of regiments and ranks greater than two, except, six.

However, a group of quantum physicists from India and Poland demonstrated that the quantum analog of this problem does have an analytic solution. The results of this research were compiled in a paper entitled "Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem", which has just been submitted for publication in Physical Review Letters, the leading physics journal.

This major finding required the use of algorithms based on quantum properties, in particular, the superposition of states. In simple terms, a superposition of states would be, for example, states consisting of a particular combination of rank and regiment for each officer. Or in the example of geometric shapes and colors, a quantum officer would be a state consisting of a combination of, for instance, a blue square and a yellow star, weighted by coefficients that are numerical values which dictate the weight or proportion of one category, with respect to another. Such coefficients are formally known as amplitudes.

The algorithm makes possible combinations and calculates amplitudes until it reaches convergence, at which point it has shown all possible options. The results were striking, for two main reasons. First, because the solution that was found was with states that were intertwined or entangled (i.e., that cannot be decomposed into separated categories), and the degree of entanglement was maximal, a characteristic that is very difficult to achieve in quantum states. In other words, this tool made it possible to find maximally entangled states in a very systematic way. Entanglement is the key feature for quantum computation, as it shields quantum states from being corrupted. Therefore, maximally entangled states are the best shield a quantum state could have.

Secondly, researchers found that the golden ratio determines all the amplitudes appearing in the maximum entanglement solution, which is why that state was named the golden maximum entanglement amplitude state.

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Curiously, the ratio of the coefficients that the algorithm landed on was Φ, or 1.618…, the famous golden ratio." Daniel Garisto, for Quanta Magazine

To visualize the solution/entanglement between the Euler quantum officers, we can observe the figure below. The position of each officer; i-th row and j-th column, is presented in the corresponding row and column in the matrix. The ranks and regiments of the officers are represented in the form of cards with corresponding ranks and suits. The grade of each officer is in a superposition of four basic grades: two ranks, and two regiments. The size of the letter of each rank corresponds to the magnitude of the amplitude of the related element. A classical solution of Euler's problem would correspond to the matrix with only one card in each box.

^{Visualization of the solution/entanglement between the quantum Euler officials. Figure taken from preprint.}

We note that the structures that arise in the matrix, for example, the aces, are entangled with the kings because they only appear with the kings, the queens only with the jacks and the 10's only with the 9's. That is, they are only and always intertwined with the immediate neighbors. In addition, the colors of the cards are not interlocked or entangled with each other. Therefore, the officers are grouped into nine sets with four elements, each of which shares the same pairs of figures and card colors.

_{RSF in perspective:}

This result is remarkable as it means that maximal entanglement originated or made possible this solution.

There are many other spectacular features worth mentioning, such as the unexpected appearance of the golden ratio (1,618....). Nassim Haramein's forthcoming paper, entitled "Scale-invariant unification of fields, forces and particles in the quantum vacuum plasma", shows how the golden ratio emerges from Haramein's generalized holographic model. This work will come hand in hand with Module 8, "The Universal Scaling Law", and once they have been published, we will be able to delve into more details about this extraordinary solution of the golden maximum entanglement amplitude, and discuss further its implications.

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