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

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 plus or minus1, sqrt2/sqrt 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.