Real four-quhex absolutely maximally entangled state

Problem

Does there exist an \(\operatorname{AME}(4,6)\) state whose coefficients are real in a product basis? Fix \([6]=\{0,\ldots,5\}\) and ask for a normalized state

\begin{equation} |\psi\rangle =\sum_{i,j,k,l\in[6]}T_{ijkl}|i\rangle_A|j\rangle_B |k\rangle_C|l\rangle_D, \qquad T_{ijkl}\in\mathbb{R}, \qquad \sum_{i,j,k,l\in[6]}T_{ijkl}^{2}=1 . \tag{1} \end{equation}

The state in (1) must have maximally mixed reductions on every pair of parties: for each \(S\subseteq\{A,B,C,D\}\) with \(|S|=2\),

\begin{equation} \rho_S:=\operatorname{Tr}_{S^{c}}|\psi\rangle\!\langle\psi| =\frac{I_{36}}{36} . \tag{2} \end{equation}

Equivalently, define the \(36\times36\) flattening \(U\) and its reshuffling \(U^R\) and second-factor partial transpose \(U^\Gamma\) by

\begin{equation} U_{(i,j),(k,l)}:=6T_{ijkl}, \qquad (U^R)_{(i,k),(j,l)}:=U_{(i,j),(k,l)}, \qquad (U^\Gamma)_{(i,j),(k,l)}:=U_{(i,l),(k,j)} . \tag{3} \end{equation}

Here the three matrices in (3) are the coefficient flattenings for the bipartitions \(AB|CD\), \(AC|BD\), and \(AD|CB\), up to the displayed ordering of tensor factors. Consequently, (2) is equivalent to the orthogonal \(2\)-unitarity conditions

\begin{equation} U\in O(36), \qquad U^R\in O(36), \qquad U^\Gamma\in O(36) . \tag{4} \end{equation}

Thus the problem is also to construct an orthogonal \(2\)-unitary matrix of order \(36\), or prove that none exists.

Source

Rajchel-Mieldzioć et al. explicitly ask whether a real \(\operatorname{AME}(4,6)\) state, equivalently an orthogonal \(2\)-unitary matrix of order \(36\), exists [RBR+26].

Progress

Rather, Burchardt, Bruzda, Rajchel-Mieldzioć, Lakshminarayan, and Życzkowski constructed an exact \(\operatorname{AME}(4,6)\) state, equivalently a complex \(2\)-unitary matrix of order \(36\). Its entries use nontrivial twentieth-root phases. Their numerical searches found high-entangling-power orthogonal matrices but no orthogonal \(2\)-unitary, leading them to conjecture nonexistence in \(O(36)\); this is numerical evidence, not a proof [RBB+22].
Cha proved, for the generalized-Pauli stabilizer convention used in that work, that no stabilizer \(\operatorname{AME}(4,6)\) state exists. Hence any real solution of (2) must lie outside that stabilizer class, but the theorem does not exclude real nonstabilizer states [Cha26].

Comment

Item T2 of [RBR+26] explicitly retains the real \(\operatorname{AME}(4,6)\) problem and its orthogonal \(2\)-unitary formulation. The precise unresolved gap is whether the three simultaneous orthogonality conditions in (4) have any real solution; neither the known complex constructions nor the stabilizer obstruction answers this question.

References

[RBB+22]: S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć, A. Lakshminarayan, and K. Życzkowski, “Thirty-six Entangled Officers of Euler: Quantum Solution to a Classically Impossible Problem,” Physical Review Letters 128, 080507 (2022).DOI arXiv
[Cha26]: H. Cha, “Non-existence of Stabilizer Absolutely Maximally Entangled States across Infinitely Many Configurations,” arXiv:2603.13442v2 [quant-ph] (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

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