Existence of mixed dimensional AME states

This table follows the convention that a mixed-dimenensional (heterogeneous) pure state is absolutely maximally entangled (AME), if maximal entanglement is present over the given bipartition. A second convention (below) asks for maximal mixed marginals on all subsystems of size \(n/2\).

Generic constructions

[BZ25], Theorem 9: If \(|\psi_1\rangle, \dots, |\psi_r\rangle\) is an orthonormal basis of a pure \((\!( D_1, \dots, D_n), K, \lceil r D_1 \dots D_n \rceil )\!)\)] code then there is a mixed-dimensional AME with dimensions \((r, D_1, \dots, D_n)\) for all \(r \leq K\).

References

[HESG18] Felix Huber, Christopher Eltschka, Jens Siewert, Otfried Gühne, Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity, J. Phys. A: Math. Theor. 51 175301 (2018), arXiv:1708.06298

[BZ25] Simeon Ball, Raven Zhang, “Error-correcting codes and absolutely maximally entangled states for mixed dimensional Hilbert spaces”, arXiv:2510.17231

Maximally mixed convention

Here the requirement is that all subsystems of size \(\lfloor \tfrac{n}{2}\rfloor\) is maximally mixed. [GBZ16] found a \(2 \times 2 \times 3\) state.

[GBZ16] Dardo Goyeneche, Jakub Bielawski, Karol Życzkowski, Multipartite entanglement in heterogeneous systems, Physical Review A 94, 012346 (2016), arXiv:1602.08064