A Latin square can be extended by replacing its symbols with probabilities or quantum states. We relate these two extensions through explicit examples, from classical and fractional Latin squares to quantum Latin squares and a real solution of Euler's quantum 36 officers problem. A simultaneous cyclic-shift ansatz on a four-torus of tensor coefficients gives a real absolutely maximally entangled (AME) state of four six-level systems. A local orthogonal change of basis yields a representative with 96 nonzero amplitudes. Two single-party arrays obtained from the corresponding quantum orthogonal Latin square yield fractional Latin squares. Signs matter: comparison with the independent real construction of Bruzda and its sign variant distinguishes three local-unitary classes, two of which have identical probabilities. We extend this probability representation to existing seven- and nine-party AME states with six levels each. Their amplitude matrices reveal 48 complex Hadamard blocks of order $3^3$ and six of order $6^4$, respectively. These constructions separate what can be read from a probability picture from the coherent conditions responsible for maximal entanglement.
A Latin square can be extended by replacing its symbols with probabilities or quantum states. We relate these two extensions through explicit examples, from classical and fractional Latin squares to quantum Latin squares and a real solution of Euler's quantum 36 officers problem. A simultaneous cyclic-shift ansatz on a four-torus of tensor coefficients gives a real absolutely maximally entangled (AME) state of four six-level systems. A local orthogonal change of basis yields a representative with 96 nonzero amplitudes. Two single-party arrays obtained from the corresponding quantum orthogonal Latin square yield fractional Latin squares. Signs matter: comparison with the independent real construction of Bruzda and its sign variant distinguishes three local-unitary classes, two of which have identical probabilities. We extend this probability representation to existing seven- and nine-party AME states with six levels each. Their amplitude matrices reveal 48 complex Hadamard blocks of order 33 and six of order 64, respectively. These constructions separate what can be read from a probability picture from the coherent conditions responsible for maximal entanglement.
The negative solution to the famous problem of 36 officers of Euler implies that there are no two orthogonal Latin squares of order six. We show that the problem has a solution, provided the officers are entangled, and construct orthogonal quantum Latin squares of this size. As a consequence, we find an example of the long-elusive Absolutely Maximally Entangled state AME(4,6) of four subsystems with six levels each, equivalently a 2-unitary matrix of size 36, which maximizes the entangling power among all bipartite unitary gates of this dimension, or a perfect tensor with four indices, each running from one to six. This special state deserves the appellation golden AME state, as the golden ratio appears prominently in its elements. This result allows us to construct a pure nonadditive quhex quantum error detection code ((3,6,2))_{6}, which saturates the Singleton bound and allows one to encode a six-level state into a triplet of such states.
Absolutely maximally entangled states of four six-level systems, AME(4,6), are the quantum solutions of Euler's problem of the thirty-six officers. Until recently all reported solutions were complex, and it was conjectured that no real orthogonal 2-unitary matrix of order 36 exists. We give real AME(4,6) states with 60 non-zero coefficients in {±1,2}/72. Of their thirty-six officers, 12 are classical, as some must be whenever fewer than 72 coefficients are non-zero. They lie in the −1 eigenspace of an order-4 local symmetry, and on their support the AME conditions become linear. We derive two closed-form families of AME(4,6) states that meet at one of these states: a 10-parameter family that is supported on 72 basis states, and an 11-parameter family on 60 basis states. At each of its two ends the first family loses 12 coefficients, and the AME states on the remaining 60 basis states form an 11-parameter family: the second family at one end, and an analogous family at the other. These two 60-term families contain 4096 real AME states up to diagonal local unitaries, 1024 of which are limits of the first family. They fall into exactly 20 classes under local unitaries and permutations of the parties, and we certify their separation with interval arithmetic.
There exist pairs of orthogonal Latin squares of any order n except if n=2 or n=6 [Bose et al., Further results on the construction of mutually orthogonal latin squares and the falsity of Euler's conjecture, Can. J. Math. 12, 189 (1960)CJMAAB0008-414X10.4153/CJM-1960-016-5]. In particular, the problem of Euler's thirty-six officers does not have a solution. However, it has a "quantum solution": there exist so-called entangled quantum Latin squares of order 6 [Rather et al., Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem, Phys. Rev. Lett. 128, 080507 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.080507]. We prove that mutually orthogonal quantum Latin squares of order 6 do not exist if entanglement is not allowed.
Geomagic Square Representation of a Real AME(4,6) State
Seeking a pictorial representation of four-party absolutely maximally entangled (AME) states of local size six, we independently construct a real AME(4,6) state from a table of signed Pauli matrices. Its two-unitary matrix has 60 nonzero entries, equal to 1 or ±1/2. A locally equivalent representative yields a genuine geomagic square, where every row, column and main diagonal tiles a hexagon and all 36 pieces are pairwise non-congruent. Colors, shapes and hatching retain the full tensor, allowing its reconstruction from the picture.
The classical combinatorial problem of 36 officers has no solution, as there are no Graeco-Latin squares of order six. The situation changes if one works in a quantum setup and allows for superpositions of classical objects and admits entangled states. We analyze the recently found solution to the quantum version of the Euler's problem from a geometric point of view. The notion of a non-displaceable manifold embedded in a larger space is recalled. This property implies that any two copies of such a manifold, like two great circles on a sphere, do intersect. Existence of a quantum Graeco-Latin square of size six, equivalent to a maximally entangled state of four subsystems with d=6 levels each, implies that three copies of the manifold U(36)/U(1) of maximally entangled states of the 36×36 system, embedded in the complex projective space CP36×36−1, do intersect simultaneously at a certain point.