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 \(n/2\).
| Existence | dimensions | Reference |
| Y | \(2 \times 2 \times 3\) | [GB26] |
| Y | \(2 \times 3 \times 3\) | [GB26] |
| Y | \(2 \times 3 \times 4\) | [GB26] |
| N | \(2 \times 3 \times 5\) | [H26] |
| Y | \(2 \times 4 \times 4\) | [H26] |
| Y | \(2 \times 4 \times 6\) | [H26] |
| Y | \(3 \times 3 \times 4\) | [GB26] |
| Y | \(3 \times 4 \times 4\) | [GB26] |
| N | \(2 \times 2 \times 2 \times 3\) | [HESG18] |
| Y | \(2 \times 2 \times 2 \times 4\) | [H26] |
| N | \(2 \times 2 \times 3 \times 3\) | [HESG18] |
| Y | \(2 \times 3 \times 3 \times 3\) | [HESG18] |
| Y | \(2 \times 5 \times 5 \times 5\) | [BZ25] |
| Y | \(3 \times 5 \times 5 \times 5\) | [BZ25] |
| Y | \(4 \times 5 \times 5 \times 5\) | [BZ25] |
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 \(r \leq K\).
No-go
Let \(p\neq q\). Then \((p,q, pq-1)\) does not exist. [H26]
References
[GB26] David González-Lociga and Simeon Ball, The mixed-dimensional quantum MacWilliams identity: bounds for codes and absolutely maximally entangled states in heterogeneous systems, arXiv:2604.25790
[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
[H26] Felix Huber, In preparation.
Maximally mixed convention
Another convention is not that of maximal entanglement across bipartitions, but maximally mixed marginals (these conditions differ on mixed-dimensional systems). In the maximally mixed convention, the requirement is that all subsystems of size \(\lfloor \tfrac{n}{2}\rfloor\) are 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