Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.18193

On Non-Existence of Absolutely Maximally Entangled Canonical Graph States in Even Local Dimensions

Jakub Wójcik, Owidiusz Makuta, Wojciech Bruzda, Remigiusz Augusiak

quant-phSubmitted Mar 18, 2026• Announced Mar 20, 2026• 0 citations• Revised Sep 28, 2026 (v3)
v1 · Mar 18, 2026v2 · Apr 27, 2026v3 · Sep 28, 2026
arXivHTMLPDF

Abstract

We demonstrate that absolutely maximally entangled (AME) states consisting of $N=4n$ qudits with $n\in\{1,2,3,\ldots\}$, each of even local dimension $d$, cannot be realized as canonical graph states over the ring $\mathbb{Z}_d$. 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 $d\equiv 2\pmod 4$ 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 $\mathbb{Z}_{d}$ graph states and stabilizer constructions over prime-power factors. We also discuss a composite construction of mixed $k$-uniform states. For $(N,d)=(4,6)$, the construction yields a rank-two mixed $2$-uniform stabilizer state of purity $1/2$, which is optimal among normalized stabilizer projectors.

Comments
5 pages

Related papers

Non-existence of stabilizer absolutely maximally entangled states across infinitely many configurations

Hyunho Cha

We prove a general reduction theorem for stabilizer absolutely maximally entangled states in composite local dimension. If a stabilizer $\mathrm{AME}(n,D)$ state exists and $D=\prod_{i=1}^m q_i$ is the prime-power factorization of $D$, then for every nonempty subset of factors there exists a stabilizer $\mathrm{AME}\bigl(n,\prod_{i\in M} q_i\bigr)$ state. Thus any obstruction at a prime-power factor immediately obstructs stabilizer AME states in the composite dimension.

quant-pharXiv:2603.13442•Mar 16, 2026•0 citations

Complete Existence Classification of Seven-Partite Absolutely Maximally Entangled States

Fei Shi, Xiande Zhang, Qi Zhao +1 more

We prove that an absolutely maximally entangled state of seven qudits exists if and only if the local dimension satisfies $d\geq 3$. Prior to this work, to the best of our knowledge, $\text{AME}(7,d)$ states were known to exist only when $d$ is a prime power other than $2$, or when $d$ can be expressed as a product of dimensions for which existence was already known. Since it has been proved that no $\text{AME}(7,2)$ state exists, it remains to establish existence for all $d\geq 3$. We construct cyclic quadratic-phase states for every odd local dimension and develop a coupled binary--odd-dimensional construction for every dimension congruent to $2$ modulo $4$. Together with the known power-of-two cases and the product property of AME states, these constructions cover every local dimension $d\geq 3$.

quant-pharXiv:2608.01011•Aug 4, 2026•0 citations

General stabilizer approach for constructing highly entangled graph states

Zahra Raissi, Adam Burchardt, Barnes, E. (Edwin)

Highly entangled multipartite states such as k-uniform (k-UNI) and absolutely maximally entangled (AME) states serve as critical resources in quantum networking and other quantum information applications. However, there does not yet exist a complete classification of such states, and much remains unknown about their entanglement structure. Here we substantially broaden the class of known k-UNI and AME states by introducing a method for explicitly constructing such states that combines classical error correcting codes and qudit graph states. This method in fact constitutes a general recipe for obtaining multipartitite entangled states from classical codes. Furthermore, we show that at least for a large subset of the class of k-UNI states that we present, the states are inequivalent under stochastic local operations and classical communication. This subset is defined by an iterative procedure for constructing a hierarchy of k-UNI graph states.

•Dec 19, 2022•12 citations

Quantum circuits for high-dimensional absolutely maximally entangled states

Berta Casas, Grzegorz Rajchel-Mieldzioć, Suhail Ahmad Rather +4 more

Abstract Absolutely maximally entangled (AME) states of multipartite quantum systems exhibit maximal entanglement across all possible bipartitions. These states lead to teleportation protocols that surpass standard teleportation schemes, determine quantum error correction codes and can be used to test performance of current term quantum processors. Several AME states can be constructed from graph states using minimal quantum resources. However, there exist other constructions that depart from the stabilizer formalism. In this work, we present explicit quantum circuits to generate exemplary non-stabilizer AME states of four subsystems with four, six, and eight levels each and analyze their capabilities to perform quantum information tasks.

arXiv:2504.05394•Quantum Science and Technology•Feb 17, 2026•1 citation

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

Mykhailo Hontarenko, Karol Życzkowski

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.

quant-pharXiv:2609.11796•Sep 11, 2026•0 citations

On the Existence of Absolutely Maximally Entangled States of Minimal Support II

Antonio Bernal Serrano

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 $d$ 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, $d=6$. Results for other values of $d$ are also given.

arXiv:1807.00218•Quantum Physics Letters•Apr 1, 2019•3 citations
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.
We prove a general reduction theorem for stabilizer absolutely maximally entangled states in composite local dimension. If a stabilizer AME(n,D)\mathrm{AME}(n,D)AME(n,D) state exists and D=∏i=1mqiD=\prod_{i=1}^m q_iD=∏i=1m​qi​ is the prime-power factorization of DDD, then for every nonempty subset of factors there exists a stabilizer AME(n,∏i∈Mqi)\mathrm{AME}\bigl(n,\prod_{i\in M} q_i\bigr)AME(n,∏i∈M​qi​) state. Thus any obstruction at a prime-power factor immediately obstructs stabilizer AME states in the composite dimension.
We prove that an absolutely maximally entangled state of seven qudits exists if and only if the local dimension satisfies d≥3d\geq 3d≥3. Prior to this work, to the best of our knowledge, AME(7,d)\text{AME}(7,d)AME(7,d) states were known to exist only when ddd is a prime power other than 222, or when ddd can be expressed as a product of dimensions for which existence was already known. Since it has been proved that no AME(7,2)\text{AME}(7,2)AME(7,2) state exists, it remains to establish existence for all d≥3d\geq 3d≥3. We construct cyclic quadratic-phase states for every odd local dimension and develop a coupled binary--odd-dimensional construction for every dimension congruent to 222 modulo 444. Together with the known power-of-two cases and the product property of AME states, these constructions cover every local dimension d≥3d\geq 3d≥3.
Highly entangled multipartite states such as k-uniform (k-UNI) and absolutely maximally entangled (AME) states serve as critical resources in quantum networking and other quantum information applications. However, there does not yet exist a complete classification of such states, and much remains unknown about their entanglement structure. Here we substantially broaden the class of known k-UNI and AME states by introducing a method for explicitly constructing such states that combines classical error correcting codes and qudit graph states. This method in fact constitutes a general recipe for obtaining multipartitite entangled states from classical codes. Furthermore, we show that at least for a large subset of the class of k-UNI states that we present, the states are inequivalent under stochastic local operations and classical communication. This subset is defined by an iterative procedure for constructing a hierarchy of k-UNI graph states.
Abstract Absolutely maximally entangled (AME) states of multipartite quantum systems exhibit maximal entanglement across all possible bipartitions. These states lead to teleportation protocols that surpass standard teleportation schemes, determine quantum error correction codes and can be used to test performance of current term quantum processors. Several AME states can be constructed from graph states using minimal quantum resources. However, there exist other constructions that depart from the stabilizer formalism. In this work, we present explicit quantum circuits to generate exemplary non-stabilizer AME states of four subsystems with four, six, and eight levels each and analyze their capabilities to perform quantum information tasks.
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 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.