arXiv:2603.18193

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

Jakub Wójcik, Owidiusz Makuta, Wojciech Bruzda, Remigiusz Augusiak

quant-phSubmitted Mar 18, 2026• Announced Mar 20, 2026• 0 citations

Abstract

We demonstrate that absolutely maximally entangled (AME) states consisting of $N=4k$ qudits with $k\in\mathbb{N}_+$, each of even local dimension, cannot be realized as graph states. 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. In particular, this study provides an independent solution of the recently discussed case of the AME state of four quhexes and clarifies its characterization within the stabilizer formalism, complementing the results of Cha [arXiv:2603.13442].

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arXiv:2603.18193

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

Jakub Wójcik, Owidiusz Makuta, Wojciech Bruzda, Remigiusz Augusiak

quant-phSubmitted Mar 18, 2026• Announced Mar 20, 2026• 0 citations

arXiv HTML PDF

Abstract

We demonstrate that absolutely maximally entangled (AME) states consisting of

N=4k

qudits with

k\in\mathbb{N}_+

, each of even local dimension, cannot be realized as graph states. 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. In particular, this study provides an independent solution of the recently discussed case of the AME state of four quhexes and clarifies its characterization within the stabilizer formalism, complementing the results of Cha [arXiv:2603.13442].

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.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:2603.13442Mar 16, 20260 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 $d\geq 3$ .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:2608.01011Aug 4, 20260 citations

Constructing absolute maximally entangled states and optimal quantum error correcting codes

Zahra Raissi, Christian Gogolin, Arnau Riera +1 more

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 $q \geq n-1$ . 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 $q \geq n-1$ 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 $[\![n,1,n/2]\!]_q$ for $n$ 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 $n = 8$ .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 $q \geq n-1$. 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 $q \geq n-1$ 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 $[\![n,1,n/2]\!]_q$ for $n$ 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 $n = 8$.

Jan 12, 20171 citation

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.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:2504.05394Quantum Science and TechnologyFeb 17, 20261 citation

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.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, 202212 citations

Absolutely Maximally Entangled States of $2q$ Parties in Every Odd Prime-Power Dimension $q$ 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.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:2609.11796Sep 11, 20260 citations

We demonstrate that absolutely maximally entangled (AME) states consisting of

N=4k

qudits with

k\in\mathbb{N}_+

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.

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

n

parties with local dimension

q

a power of a prime for all

q \geq 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 \geq 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

for

n

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 = 8

Absolutely Maximally Entangled States of

2q

Parties in Every Odd Prime-Power Dimension

q

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.