Talk 4.   Quantum designs and Absolutely Maximally Entangled (AME) states.

 An AME state of a system consisting of 2k parties is distinguished by the fact that for any splitting of the system into two parts with k subsystem each, both parties are maximally entangled. Such states, useful to construct quantum error-correction codes and teleportation schemes, are known for several systems including four systems with N=3,4,5,7,8... levels each and a six-qubit system. We show that the AME(4,6) state of four subsystems with six levels each exists and present an analytical solution, equivalent to a 2-unitary matrix of order 36 and a perfect tensor with 4 indices running from one to six [4]. Furthermore, it yields a quantum error correcting ((3,6,2))6 code and can be considered as a quantum solution of the famous 36-officers problem of Euler with entangled officers. We tend to believe this result will trigger further research in the field of quantum designs and quantum combinatorics.

[4] S. A. Rather. A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć, A. Lakshminarayan, and  K. Życzkowski, Thirty-six entangled officers of Euler and a quhex quantum error correcting code, preprint, April 2021