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arXiv:2609.11796

Absolutely Maximally Entangled States of $2q$ Parties in Every Odd Prime-Power Dimension $q$

Mykhailo Hontarenko, Karol Życzkowski

quant-phSubmitted Sep 10, 2026• Announced Sep 11, 2026• 0 citations
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Abstract

Absolutely maximally entangled (AME) states represent an extreme form of multipartite entanglement: every reduced system containing at most half of the parties is maximally mixed. These states provide perfect tensors and optimal quantum error-correcting codes, yet their existence is known only in restricted parameter regimes. For every odd prime power $q=p^e\ge3$, we construct a stabilizer $\mathrm{AME}(2q,q)$ state whose normalized one-party projection yields a stabilizer $\mathrm{AME}(2q-1,q)$ state. A closed-form $q\times q$ bordered-circulant matrix $A_q$ over $\mathbb{F}_{q^2}$ generates a Hermitian self-dual maximum distance separable (MDS) code $[2q,q,q+1]_{q^2}$, which lies outside the extended Reed-Solomon classes. In suitable bases, the amplitude tensors define normalized $q$-unitary complex Hadamard matrices of order $q^q$ with $p$th-root phases. Additional constructions yield $\mathrm{AME}(q+3,q)$ states and families at intermediate particle numbers through explicit rescalings of selected submatrices. We also provide nine explicit parent matrices and the corresponding one-party projections.

Comments
24 pages, 1 figure. Supplemental Material included

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Absolutely Maximally Entangled States of 2q2q2q Parties in Every Odd Prime-Power Dimension qqq
Absolutely maximally entangled (AME) states represent an extreme form of multipartite entanglement: every reduced system containing at most half of the parties is maximally mixed. These states provide perfect tensors and optimal quantum error-correcting codes, yet their existence is known only in restricted parameter regimes. For every odd prime power q=pe≥3q=p^e\ge3q=pe≥3, we construct a stabilizer AME(2q,q)\mathrm{AME}(2q,q)AME(2q,q) state whose normalized one-party projection yields a stabilizer AME(2q−1,q)\mathrm{AME}(2q-1,q)AME(2q−1,q) state. A closed-form q×qq\times qq×q bordered-circulant matrix AqA_qAq​ over Fq2\mathbb{F}_{q^2}Fq2​ generates a Hermitian self-dual maximum distance separable (MDS) code [2q,q,q+1]q2[2q,q,q+1]_{q^2}[2q,q,q+1]q2​, which lies outside the extended Reed-Solomon classes. In suitable bases, the amplitude tensors define normalized qqq-unitary complex Hadamard matrices of order qqq^qqq with pppth-root phases. Additional constructions yield AME(q+3,q)\mathrm{AME}(q+3,q)AME(q+3,q) states and families at intermediate particle numbers through explicit rescalings of selected submatrices. We also provide nine explicit parent matrices and the corresponding one-party projections.
Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of nnn parties with local dimension qqq a power of a prime for all q≥n−1q \geq n-1q≥n−1. Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every q≥n−1q \geq n-1q≥n−1 prime we show how to construct QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes [ ⁣[n,1,n/2] ⁣]q[\![n,1,n/2]\!]_q[[n,1,n/2]]q​ for nnn even, saturating the quantum Singleton bound. We show that our conjecture is equivalent to the existence of an operator whose support cannot be decreased by multiplying it with stabilizer products and explicitly construct the codes up to n=8n = 8n=8.
We give explicit Hermitian self-dual maximum distance separable codes with parameters [12,6,7]25[12,6,7]_{25}[12,6,7]25​, [18,9,10]121[18,9,10]_{121}[18,9,10]121​, and [18,9,10]169[18,9,10]_{169}[18,9,10]169​. The stabilizer construction proves the existence of AME(12,5){\rm AME}(12,5)AME(12,5), AME(18,11){\rm AME}(18,11)AME(18,11), and AME(18,13){\rm AME}(18,13)AME(18,13) states; one-party projection also gives AME(17,11){\rm AME}(17,11)AME(17,11) and AME(17,13){\rm AME}(17,13)AME(17,13). The first code was found by a direct search. A regular Z32\mathbb{Z}_3^2Z32​ coordinate orbit of its automorphism group suggested a group-circulant form that reduces each length-eighteen construction to a nine-entry kernel. The printed matrices are certified by exact Hermitian products and complete square-minor enumeration.
Abstract Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of n parties with local dimension q a power of a prime for all q ⩾ n − 1 . Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every q ⩾ n − 1 prime, we show how to construct stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes [ [ n , 1 , n / 2 ] ] q for n even with the highest distance allowed by the quantum Singleton bound.
We demonstrate that absolutely maximally entangled (AME) states consisting of N=4nN=4nN=4n qudits with n∈{1,2,3,…}n\in\{1,2,3,\ldots\}n∈{1,2,3,…}, each of even local dimension ddd, cannot be realized as canonical graph states over the ring Zd\mathbb{Z}_dZd​. This result imposes strong constraints on AME states in composite local dimensions and characterizes the limitations of graph-state constructions for highly entangled multipartite quantum systems. Furthermore, for d≡2(mod4)d\equiv 2\pmod 4d≡2(mod4) this obstruction, combined with the prime-power decomposition of stabilizer states, excludes all pure stabilizer AME states (including four quhexes), while clarifying the distinction between canonical Zd\mathbb{Z}_{d}Zd​ graph states and stabilizer constructions over prime-power factors. We also discuss a composite construction of mixed kkk-uniform states. For (N,d)=(4,6)(N,d)=(4,6)(N,d)=(4,6), the construction yields a rank-two mixed 222-uniform stabilizer state of purity 1/21/21/2, which is optimal among normalized stabilizer projectors.
Absolutely maximally entangled, AME, states are pure multipartite states that give rise to the maximally mixed states when half or more of the parties are traced out. AME states find applications in fields like teleportation or quantum secret sharing, and the problem of finding conditions on their existence has been considered in a number of papers. We consider here AME states of minimal support, that are simpler to analyse. An equivalence with coding theory gives a sufficient condition for their existence, that the number of sites be equal to the local dimension plus one, when the local dimension ddd is a power of a prime number. In this paper, an equivalence with latin hypercubes is used to prove that the above sufficient condition fails in the first case in which the local dimension is not a prime power, d=6d=6d=6. Results for other values of ddd are also given.
Absolutely maximally entangled (AME) states and, more generally, kkk-uniform states in $(\C^q)^{\otimes n}$ are central objects in multipartite entanglement theory, with applications to quantum secret sharing, quantum masking, and quantum error correction. In the extremal case k=⌊n/2⌋k=\lfloor n/2\rfloork=⌊n/2⌋, Scott (2004) proved a sharp nonexistence bound showing that AME states cannot exist once the number of parties nnn exceeds a threshold of order 2q22q^{2}2q2 (with a parity dependence on nnn), where qqq is the local dimension. Recently, Ning et al.\ studied \emph{defective} AME states (i.e., k=⌊n/2⌋−lk=\lfloor n/2\rfloor-lk=⌊n/2⌋−l with l>0l>0l>0), gave explicit Scott-type bounds for defects l=1,2l=1,2l=1,2 and conjectured a general (2l+2)q2+o(q2)(2l+2)q^{2}+o(q^{2})(2l+2)q2+o(q2) behavior. In this paper, we solve this conjecture and establish a fully explicit Scott-type upper bound for AME states with arbitrary defect l≥0l\ge 0l≥0, yielding Scott's bound for l=0l=0l=0 and Ning et al.'s bounds for l=1,2l=1,2l=1,2 as special cases. Equivalently, this gives nonexistence bounds for one-dimensional pure quantum error-correcting codes near the quantum Singleton regime. The proof uses a truncated MacWilliams linear-programming system and an explicit infeasibility certificate. As a direct application, we derive explicit asymptotic upper bounds on k/nk/nk/n for fixed local dimension qqq, improving the implicit upper bounds given by Ning et al.