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 $\pm 1/\sqrt{2}$. 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.
Comments
6 pages, 2 figures, 1 table. Scripts and data: https://github.com/matrix-toolbox/AME_4_6
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.
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.
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.
Ordering and classifying multipartite quantum states by their entanglement content remains an open problem. One class of highly entangled states, useful in quantum information protocols, the absolutely maximally entangled (AME) ones, are especially hard to compare as all their subsystems are maximally random. While it is well known that there is no AME state of four qubits, many analytical examples and numerically generated ensembles of four-qutrit AME states are known. However, we prove the surprising result that there is truly only one AME state of four qutrits up to local unitary equivalence. In contrast, for larger local dimensions, the number of local unitary classes of AME states is shown to be infinite. Of special interest is the case of local dimension 6, where it was established recently that a four-party AME state does exist, providing a quantum solution to the classically impossible Euler problem of 36 officers. Based on this, an infinity of quantum solutions are constructed, and we prove that these are not equivalent. The methods developed can be usefully generalized to multipartite states of any number of particles.
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 (AME) states of k qudits (also known as perfect tensors) are quantum states that have maximal entanglement for all possible bipartitions of the sites/parties. We consider the problem of whether such states can be decomposed into a tensor network with a small number of tensors, such that all physical and all auxiliary spaces have the same dimension D . We find that certain AME states with k=6 can be decomposed into a network with only three 4-leg tensors; we provide concrete solutions for local dimension D=5 and higher. Our result implies that certain AME states with six parties can be created with only three two-site unitaries from a product state of three Bell pairs, or equivalently, with six two-site unitaries acting on a product state on six qudits. We also consider the problem for k=8 , where we find similar tensor network decompositions with six 4-leg tensors.
We demonstrate that absolutely maximally entangled (AME) states consisting of N=4n qudits with n∈{1,2,3,…}, each of even local dimension d, cannot be realized as canonical graph states over the ring Zd. This result imposes strong constraints on AME states in composite local dimensions and characterizes the limitations of graph-state constructions for highly entangled multipartite quantum systems. Furthermore, for d≡2(mod4) this obstruction, combined with the prime-power decomposition of stabilizer states, excludes all pure stabilizer AME states (including four quhexes), while clarifying the distinction between canonical Zd graph states and stabilizer constructions over prime-power factors. We also discuss a composite construction of mixed k-uniform states. For (N,d)=(4,6), the construction yields a rank-two mixed 2-uniform stabilizer state of purity 1/2, which is optimal among normalized stabilizer projectors.