Finite nontrivial LU moduli of AME states

Problem

Do there exist integers \(N,d\geq 2\) for which the absolutely maximally entangled states of \(N\) qudits of local dimension \(d\) form finitely many, but more than one, local-unitary equivalence classes? A normalized vector \(|\psi\rangle\in(\mathbb{C}^{d})^{\otimes N}\) is an \(\operatorname{AME}(N,d)\) state when every subsystem of at most half the parties is maximally mixed:

\begin{equation} \operatorname{Tr}_{S^{c}}\!\left(|\psi\rangle\!\langle\psi|\right) =\frac{I_{d^{|S|}}}{d^{|S|}} \quad\text{for every }S\subseteq\{1,\ldots,N\} \text{ with }|S|\leq\left\lfloor\frac{N}{2}\right\rfloor . \tag{1} \end{equation}

For normalized states satisfying Eq. (1), define local-unitary equivalence by

\begin{equation} |\psi\rangle\sim_{\mathrm{LU}}|\phi\rangle \quad\Longleftrightarrow\quad |\psi\rangle=(U_1\otimes\cdots\otimes U_N)|\phi\rangle \quad\text{for some }U_1,\ldots,U_N\in U(d). \tag{2} \end{equation}

If \(\mathcal{A}_{N,d}\) denotes the set specified by Eq. (1), determine whether the quotient under Eq. (2) can satisfy

\begin{equation} \mathfrak{M}_{N,d}:=\mathcal{A}_{N,d}/\!\sim_{\mathrm{LU}}, \qquad 1<\lvert\mathfrak{M}_{N,d}\rvert<\infty . \tag{3} \end{equation}

Source

Rajchel-Mieldzioć et al. explicitly ask whether an AME family can have a finite number of local-unitary equivalence classes greater than one [RBR+26].

Progress

For four parties, all \(\operatorname{AME}(4,3)\) states lie in one LU class, whereas \(\operatorname{AME}(4,d)\) has infinitely many LU classes for every \(d\geq 4\) [RRKL23]. The latter theorem includes infinitely many inequivalent \(\operatorname{AME}(4,6)\) states, so neither side of this classification realizes Eq. (3).
Tan completely classifies five-qubit AME states: every such state is LU equivalent to a point of the unique \(((5,2,3))\) code \(\mathcal C\), and two points of \(\mathcal C\) are equivalent exactly when related by its finite binary-tetrahedral transversal group [Tan26]. Because the projective state space of the two-dimensional code is a continuum while this group is finite, \(\lvert\mathfrak{M}_{5,2}\rvert\) is infinite, not finite and nontrivial.
The 2026 AME review records the known one-class and infinite-class regimes and explicitly leaves Eq. (3) as open problem T5 [RBR+26].

Comment

The remaining task is either to exhibit an AME parameter pair whose full LU quotient is a discrete non-singleton set, or to prove that every nonempty \(\mathfrak{M}_{N,d}\) is a singleton or infinite. Classifying only a selected construction family would not determine the full quotient in Eq. (3).

References

[RRKL23]: S. A. Rather, N. Ramadas, V. Kodiyalam, and A. Lakshminarayan, “Absolutely Maximally Entangled State Equivalence and the Construction of Infinite Quantum Solutions to the Problem of 36 Officers of Euler,” Physical Review A 108, 032412 (2023).DOI arXiv
[Tan26]: I. Tan, “Classification of Five-Qubit Absolutely Maximally Entangled States,” arXiv:2507.02185v4 (2026).arXiv
[RBR+26]: G. Rajchel-Mieldzioć, R. Bistroń, A. Rico, A. Lakshminarayan, and K. Życzkowski, “Absolutely Maximally Entangled Pure States of Multipartite Quantum Systems,” Reports on Progress in Physics 89, 057601 (2026).DOI arXiv

Finite nontrivial LU moduli of AME states

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