Classification of five-qubit absolutely maximally entangled states

Ian Tan

arxiv: 2507.02185 · v4 · pith:VR7PJOPDnew · submitted 2025-07-02 · 🪐 quant-ph · math.RT

Classification of five-qubit absolutely maximally entangled states

Ian Tan This is my paper

Pith reviewed 2026-05-21 23:21 UTC · model grok-4.3

classification 🪐 quant-ph math.RT

keywords absolutely maximally entangled statesfive qubitsquantum error-correcting codeslocal unitary equivalencetransversal gatesVinberg theorygraded Lie algebrasinvariant polynomials

0 comments

The pith

Every 5-qubit AME state is locally equivalent to one inside the unique ((5,2,3)) quantum error-correcting code, with equivalences given exactly by its transversal gates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies all local unitary equivalence classes of five-qubit absolutely maximally entangled states by showing they correspond to states inside one specific quantum error-correcting code. Equivalence between any two such states holds exactly when a transversal gate from that code maps one to the other. The proof relies on mapping four-qubit states into an eight-by-eight skew-symmetric matrix Lie algebra and then using graded Lie algebra results to find all orbits. Three invariant polynomials are given to separate the resulting classes.

Core claim

We show that every 5-qubit AME state is equivalent to a state within the unique ((5,2,3)) quantum error-correcting code C, and that two such states are equivalent if and only if they are related by the action of a transversal gate of C. We exhibit a set of three invariant polynomials that separates these equivalence classes. The classification proceeds by embedding the 4-qubit state space into the Lie algebra of 8×8 skew-symmetric matrices and applying Vinberg's theory of graded Lie algebras.

What carries the argument

Embedding of the 4-qubit state space into the Lie algebra of 8×8 skew-symmetric matrices that enables application of Vinberg's theory of graded Lie algebras to classify local unitary orbits.

If this is right

Every even n at least 6 admits a 3-uniform n-qubit state.
The local symmetries of the 6-qubit AME state relate to the transversal gates of both the ((5,2,3)) and ((4,4,2)) codes.
Every 4-qubit pure code of distance 2 is equivalent to a subspace of a ((4,4,2)) code.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This classification technique using code transversals and Lie algebra embeddings may extend to seven or more qubits by linking AME states to suitable larger codes.
The three invariant polynomials supply a concrete computational test for local equivalence of specific AME states.
The direct tie between AME states and transversal gates could inform constructions that turn entangled resources into fault-tolerant quantum gates.

Load-bearing premise

The embedding of the 4-qubit state space into the Lie algebra of 8×8 skew-symmetric matrices is faithful and complete enough that Vinberg's theory of graded Lie algebras classifies all local unitary orbits of 5-qubit AME states without missing classes or introducing extraneous equivalences.

What would settle it

A five-qubit AME state that cannot be transformed by local unitaries into any state from the ((5,2,3)) code would disprove the classification, as would two states related by a transversal gate yet separated by the three invariant polynomials.

read the original abstract

We classify the local unitary equivalence classes of absolutely maximally entangled (AME) states of five qubits. We show that every 5-qubit AME state is equivalent to a state within the unique ((5,2,3)) quantum error-correcting code $\mathcal{C}$, and that two such states are equivalent if and only if they are related by the action of a transversal gate of $\mathcal{C}$. Furthermore, we exhibit a set of three invariant polynomials that separates these equivalence classes. As auxiliary results, we construct a 3-uniform $n$-qubit state for even $n\geq 6$, determine the local symmetries of the 6-qubit AME state, and explain how these symmetries are related to the transversal gates of both the ((5,2,3)) and ((4,4,2)) codes. Additionally, we demonstrate that every 4-qubit pure code of distance 2 is equivalent to a subspace of a ((4,4,2)) code. Our approach leverages an embedding of the 4-qubit state space into the Lie algebra of $8\times 8$ skew-symmetric matrices, allowing us to apply results from Vinberg's theory of graded Lie algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper classifies all 5-qubit AME states up to local unitaries by reducing them to the ((5,2,3)) code with equivalences given exactly by its transversal gates, using an embedding into skew-symmetric matrices and Vinberg theory.

read the letter

The main result is that every 5-qubit AME state is locally equivalent to one inside the unique ((5,2,3)) code, two such states are equivalent precisely when a transversal gate of that code takes one to the other, and three explicit invariant polynomials separate the classes. This is presented as new work that organizes the local unitary orbits for this case. The approach embeds the 4-qubit state space into the Lie algebra of 8x8 skew-symmetric matrices and then applies Vinberg's graded Lie algebra results to classify the orbits. The auxiliary pieces include a construction of 3-uniform n-qubit states for even n at least 6, a determination of the local symmetries of the 6-qubit AME state and how they connect to transversal gates in both the ((5,2,3)) and ((4,4,2)) codes, plus the observation that every 4-qubit pure distance-2 code sits inside a ((4,4,2)) code. These side results stand on their own and add concrete value beyond the main classification. The potential soft spot is the embedding step itself. The whole reduction depends on that map being faithful on the relevant variety so that the induced grading matches local unitary action without missing orbits or creating false identifications. The abstract states the outcome cleanly, but the strength of the claim rests on whether the full derivations and checks against known examples confirm there are no gaps in coverage. This is for people working on multipartite entanglement and its links to quantum error correction who want an explicit reference for the 5-qubit case. A reader interested in algebraic techniques for orbit classification might also find the Vinberg application worth looking at. It deserves a serious referee because the result is specific, the invariants are given explicitly, and the auxiliary facts are reproducible even if the embedding needs careful verification.

Referee Report

2 major / 2 minor

Summary. The paper classifies the local unitary equivalence classes of five-qubit absolutely maximally entangled (AME) states. It claims that every such state is locally unitarily equivalent to a state inside the unique ((5,2,3)) quantum error-correcting code C, with two states equivalent if and only if they differ by a transversal gate of C. Three invariant polynomials are exhibited that separate the classes. Auxiliary results include construction of 3-uniform n-qubit states for even n≥6, determination of local symmetries of the 6-qubit AME state and their relation to transversal gates of the ((5,2,3)) and ((4,4,2)) codes, and a demonstration that every 4-qubit pure code of distance 2 is equivalent to a subspace of a ((4,4,2)) code. The method relies on embedding the 4-qubit state space into the Lie algebra of 8×8 skew-symmetric matrices and applying Vinberg's theory of graded Lie algebras.

Significance. If the central classification holds, the work supplies a complete and explicit description of all 5-qubit AME states up to local unitaries, reducing the problem to the well-studied structure of a single stabilizer code and its transversal gates. The explicit invariant polynomials are a practical strength for distinguishing orbits. The auxiliary results on 3-uniform states and code equivalences are of independent interest and broaden the paper's scope. The algebraic approach via Vinberg theory offers a new tool for entanglement classification that may generalize.

major comments (2)

[embedding construction and Vinberg application] The embedding of the 4-qubit state space into the Lie algebra of 8×8 skew-symmetric matrices (described in the approach paragraph of the abstract and developed in the main technical section) is load-bearing for the claim that every 5-qubit AME state reduces to one inside C. The manuscript must explicitly verify that this embedding is injective on the relevant variety, that the induced grading corresponds bijectively to local unitary actions, and that no orbits are missed or spuriously identified; without such a check the reduction to the transversal gates of C remains unconfirmed.
[invariant polynomials section] The statement that the three invariant polynomials separate all equivalence classes (main classification theorem) rests on the completeness of the Vinberg orbit classification. An explicit demonstration or computational check that these polynomials distinguish the orbits obtained from the graded Lie algebra is required to substantiate the separation claim.

minor comments (2)

[auxiliary results on symmetries] Clarify the precise relation between the local symmetries of the 6-qubit AME state and the transversal gates of both the ((5,2,3)) and ((4,4,2)) codes; a short table or diagram would improve readability.
[introduction] Ensure that all references to prior classifications of AME states or quantum codes are updated to include the most recent literature on 5-qubit entanglement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and will incorporate the requested verifications in a revised version.

read point-by-point responses

Referee: [embedding construction and Vinberg application] The embedding of the 4-qubit state space into the Lie algebra of 8×8 skew-symmetric matrices (described in the approach paragraph of the abstract and developed in the main technical section) is load-bearing for the claim that every 5-qubit AME state reduces to one inside C. The manuscript must explicitly verify that this embedding is injective on the relevant variety, that the induced grading corresponds bijectively to local unitary actions, and that no orbits are missed or spuriously identified; without such a check the reduction to the transversal gates of C remains unconfirmed.

Authors: We agree that an explicit verification strengthens the argument. In the revised manuscript we will insert a new subsection that (i) proves injectivity of the embedding on the affine variety of 5-qubit AME states by direct computation of the differential and kernel, (ii) shows that the induced grading on the Lie algebra coincides with the infinitesimal action of the local unitary group SU(2)^5, and (iii) confirms completeness by matching the dimension of the quotient space with the number of orbits obtained from the transversal gates of C. These additions will make the reduction to the ((5,2,3)) code fully rigorous. revision: yes
Referee: [invariant polynomials section] The statement that the three invariant polynomials separate all equivalence classes (main classification theorem) rests on the completeness of the Vinberg orbit classification. An explicit demonstration or computational check that these polynomials distinguish the orbits obtained from the graded Lie algebra is required to substantiate the separation claim.

Authors: The three polynomials are the fundamental invariants furnished by Vinberg theory for the graded Lie algebra in question and therefore separate orbits by construction. To supply the requested explicit check we will add a short computational appendix that evaluates the three polynomials on a complete set of orbit representatives (obtained from the graded Lie algebra) and tabulates the resulting distinct values, thereby confirming separation in practice. revision: yes

Circularity Check

0 steps flagged

No circularity; classification derives from external Vinberg theory and embedding

full rationale

The paper's central derivation embeds the 4-qubit state space into the Lie algebra of 8×8 skew-symmetric matrices and invokes Vinberg's theory of graded Lie algebras to classify local unitary orbits of 5-qubit AME states, reducing every such state to one inside the ((5,2,3)) code with equivalences given by its transversal gates. This chain relies on independent algebraic structures and external theorems rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation whose content reduces to the present paper's inputs. Auxiliary constructions (3-uniform states, local symmetries) are presented as supporting results but do not tautologically define the main classification. The approach is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard results from Lie algebra theory and quantum coding theory; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)

domain assumption The embedding of the 4-qubit state space into the Lie algebra of 8×8 skew-symmetric matrices is well-defined and respects local unitary action.
Invoked to apply Vinberg's theory to the classification problem.
standard math Vinberg's theory of graded Lie algebras classifies the relevant orbits completely for this embedding.
Background theorem used to obtain the equivalence classes.

pith-pipeline@v0.9.0 · 5730 in / 1477 out tokens · 46996 ms · 2026-05-21T23:21:29.516969+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing via so(8) topology) echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the 4-qubit state space (C²)⊗4 is the grade-1 piece of the Z2-graded complex Lie algebra so8 ≅ so×2 4 ⊕ (C²)⊗4. The code C2 embedded in so8 is a Cartan subalgebra.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and orbit embedding refines

?

refines
Relation between the paper passage and the cited Recognition theorem.

every 5-qubit AME state is equivalent to a state within the unique ((5,2,3)) quantum error-correcting code C, and two such states are equivalent if and only if they are related by the action of a transversal gate of C.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Graph restricted tensors: building blocks for holographic networks
quant-ph 2025-12 unverdicted novelty 7.0

Graph-restricted tensors generalize 1-uniform states, dual-unitary operators and AME states, with exact analytic solutions for new examples motivated by holographic lattice models.
Transversal gates of the ((3,3,2)) qutrit code and local symmetries of the absolutely maximally entangled state of four qutrits
quant-ph 2026-01 unverdicted novelty 6.0

A bijection maps LU orbits of AME states (n even) to orbits of ((n-1,D,n/2))_D codes, with the 4-qutrit AME state and ((3,3,2))_3 code both unique up to LU and their symmetry/transversal groups explicitly generated vi...

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · cited by 2 Pith papers · 1 internal anchor

[1]

Bacon, Introduction to quantum error correction (D

D. Bacon, Introduction to quantum error correction (D. A. Lidar and T. A. E. Brun, eds.), Cambridge University Press, 2013

work page 2013
[2]

44, 13407

A Borras, A R Plastino, J Batle, C Zander, M Casas, and A Plastino,Multiqubit systems: highly entangled states and entanglement distribution , Journal of Physics A: Mathematical and Theoretical 40 (2007), no. 44, 13407

work page 2007
[3]

Burchardt and Z

A. Burchardt and Z. Raissi, Stochastic local operations with classical communication of absolutely max- imally entangled states , Phys. Rev. A 102 (2020), 022413

work page 2020
[4]

Normal Forms and Tensor Ranks of Pure States of Four Qubits

O. Chterental and D. Djokovic, Normal Forms and Tensor Ranks of Pure States of Four Qubits , Linear Algebra Research Advances, 2007, pp. 133–167. doi:10.48550/arXiv.quant-ph/0612184

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.quant-ph/0612184 2007
[5]

D. A. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra , 4th ed., Springer, Cham, Switzerland, 2015

work page 2015
[6]

Derksen and G

H. Derksen and G. Kemper, Computational Invariant Theory, 1st ed., Springer-Verlag Berlin Heidelberg, New York, 2002

work page 2002
[7]

Dietrich, W

H. Dietrich, W. A de Graaf, A. Marrani, and M. Origlia, Classification of four qubit states and their stabilisers under SLOCC operations , J. Phys. A (2022)

work page 2022
[8]

D¨ ur, G

W. D¨ ur, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways , Phys. Rev. A 62 (2000), 062314

work page 2000
[9]

Englbrecht and B

M. Englbrecht and B. Kraus, Symmetries and entanglement of stabilizer states, Phys. Rev. A101 (2020), 062302

work page 2020
[10]

Gottesman, Theory of fault-tolerant quantum computation , Phys

D. Gottesman, Theory of fault-tolerant quantum computation , Phys. Rev. A 57 (1998), 127–137

work page 1998
[11]

Gour and N

G. Gour and N. R. Wallach, Necessary and sufficient conditions for local manipulation of multipartite pure quantum states , New J Phys 13 (2011), no. 7, 073013

work page 2011
[12]

, Classification of multipartite entanglement of all finite dimensionality , Phys. Rev. Lett. 111 (2013), 060502

work page 2013
[13]

Goyeneche, Z

D. Goyeneche, Z. Raissi, S. Di Martino, and K. ˙Zyczkowski, Entanglement and quantum combinatorial designs, Phys. Rev. A 97 (2018), 062326. 22 IAN TAN

work page 2018
[14]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry . Available at http://www.math.uiuc.edu/Macaulay2/

work page
[15]

Enrico Fermi

M. Hein, W. D¨ ur, E. J., R. R., M. Van den Nest, and B. H.-J., Quantum computers, algorithms and chaos, Proceedings of the International School of Physics “Enrico Fermi”, vol. 162, IOS Press, 2006

work page 2006
[16]

Helwig, Absolutely maximally entangled qudit graph states , 2013

W. Helwig, Absolutely maximally entangled qudit graph states , 2013

work page 2013
[17]

, Absolutely maximally entangled qudit graph states , 2013

work page 2013
[18]

Helwig, W

W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.-K. Lo, Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A 86 (November 2012), 052335

work page 2012
[19]

Higuchi and A

A. Higuchi and A. Sudbery, How entangled can two couples get? , Physics Letters A 273 (2000), no. 4, 213–217

work page 2000
[20]

Huber and N

F. Huber and N. Wyderka, Table of AME states . Available at http://www.tp.nt.uni-siegen.de/ +fhuber/ame.html

work page
[21]

Huber, C

F. Huber, C. Eltschka, J. Siewert, and O. G¨ uhne,Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity , Journal of Physics A: Mathematical and Theoretical 51 (2018), no. 17, 175301

work page 2018
[22]

Huber and M

F. Huber and M. Grassl, Quantum codes of maximal distance and highly entangled subspaces , Quantum 4 (202006), 284

work page
[23]

Huber, O

F. Huber, O. G¨ uhne, and J. Siewert,Absolutely maximally entangled states of seven qubits do not exist , Phys. Rev. Lett. 118 (2017), 200502

work page 2017
[24]

Luque and J.-Y

J.-G. Luque and J.-Y. Thibon, Algebraic invariants of five qubits , Journal of Physics A: Mathematical and General 39 (2005), no. 2, 371

work page 2005
[25]

Martin, O

J. Martin, O. Giraud, P. A. Braun, D. Braun, and T. Bastin, Multiqubit symmetric states with high geometric entanglement, Phys. Rev. A 81 (2010), 062347

work page 2010
[26]

Oeding and S

L. Oeding and S. Sam, Equations for the fifth secant variety of segre products of projective spaces , Experimental Mathematics 25 (2016), no. 1, 94–99

work page 2016
[27]

Oeding and I

L. Oeding and I. Tan, Four-qubit critical states , Journal of Physics A: Mathematical and Theoretical 58 (2025), no. 26, 265301

work page 2025
[28]

1, 33–57

, Tensor decompositions with applications to local unitary and stochastic local operations and classical communication equivalence of multipartite pure states , SIAM Journal on Applied Algebra and Geometry 9 (2025), no. 1, 33–57

work page 2025
[29]

E. M. Rains, Nonbinary quantum codes , IEEE Transactions on Information Theory 45 (1999), no. 6, 1827–1832

work page 1999
[30]

1, 266–271

, Quantum codes of minimum distance two, IEEE Transactions on Information Theory45 (1999), no. 1, 266–271

work page 1999
[31]

E. M. Rains, Quantum weight enumerators , IEEE Trans. Inf. Theory 44 (1996), 1388–1394

work page 1996
[32]

J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics , 3rd ed., Cambridge University Press, 2020

work page 2020
[33]

C. Spee, J. I. de Vicente, D. Sauerwein, and B. Kraus, Entangled pure state transformations via local operations assisted by finitely many rounds of classical communication, Phys. Rev. Lett. 118 (2017Jan), 040503

work page
[34]

Sturmfels, Algorithms in Invariant Theory , Springer Vienna, 2008

B. Sturmfels, Algorithms in Invariant Theory , Springer Vienna, 2008

work page 2008
[35]

´E B Vinberg, The Weyl group of a graded Lie algebra , Mathematics of the USSR-Izvestiya 10 (1976), no. 3, 463

work page 1976
[36]

N. R. Wallach, Geometric Invariant Theory: Over the Real and Complex Numbers , Springer Cham, 2017

work page 2017
[37]

W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits , Phys. Rev. Lett. 80 (1998), 2245–2248

work page 1998
[38]

Y. Zang, Z. Tian, S.-M. Fei, and H.-J. Zuo, Quantum k-uniform states from quantum orthogonal arrays , International Journal of Theoretical Physics 62 (2023)

work page 2023
[39]

D. ˇZ. ¯Dokovi´ c and A. Osterloh,On polynomial invariants of several qubits , Journal of Mathematical Physics 50 (200903), no. 3, 033509

work page

[1] [1]

Bacon, Introduction to quantum error correction (D

D. Bacon, Introduction to quantum error correction (D. A. Lidar and T. A. E. Brun, eds.), Cambridge University Press, 2013

work page 2013

[2] [2]

44, 13407

A Borras, A R Plastino, J Batle, C Zander, M Casas, and A Plastino,Multiqubit systems: highly entangled states and entanglement distribution , Journal of Physics A: Mathematical and Theoretical 40 (2007), no. 44, 13407

work page 2007

[3] [3]

Burchardt and Z

A. Burchardt and Z. Raissi, Stochastic local operations with classical communication of absolutely max- imally entangled states , Phys. Rev. A 102 (2020), 022413

work page 2020

[4] [4]

Normal Forms and Tensor Ranks of Pure States of Four Qubits

O. Chterental and D. Djokovic, Normal Forms and Tensor Ranks of Pure States of Four Qubits , Linear Algebra Research Advances, 2007, pp. 133–167. doi:10.48550/arXiv.quant-ph/0612184

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.quant-ph/0612184 2007

[5] [5]

D. A. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra , 4th ed., Springer, Cham, Switzerland, 2015

work page 2015

[6] [6]

Derksen and G

H. Derksen and G. Kemper, Computational Invariant Theory, 1st ed., Springer-Verlag Berlin Heidelberg, New York, 2002

work page 2002

[7] [7]

Dietrich, W

H. Dietrich, W. A de Graaf, A. Marrani, and M. Origlia, Classification of four qubit states and their stabilisers under SLOCC operations , J. Phys. A (2022)

work page 2022

[8] [8]

D¨ ur, G

W. D¨ ur, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways , Phys. Rev. A 62 (2000), 062314

work page 2000

[9] [9]

Englbrecht and B

M. Englbrecht and B. Kraus, Symmetries and entanglement of stabilizer states, Phys. Rev. A101 (2020), 062302

work page 2020

[10] [10]

Gottesman, Theory of fault-tolerant quantum computation , Phys

D. Gottesman, Theory of fault-tolerant quantum computation , Phys. Rev. A 57 (1998), 127–137

work page 1998

[11] [11]

Gour and N

G. Gour and N. R. Wallach, Necessary and sufficient conditions for local manipulation of multipartite pure quantum states , New J Phys 13 (2011), no. 7, 073013

work page 2011

[12] [12]

, Classification of multipartite entanglement of all finite dimensionality , Phys. Rev. Lett. 111 (2013), 060502

work page 2013

[13] [13]

Goyeneche, Z

D. Goyeneche, Z. Raissi, S. Di Martino, and K. ˙Zyczkowski, Entanglement and quantum combinatorial designs, Phys. Rev. A 97 (2018), 062326. 22 IAN TAN

work page 2018

[14] [14]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry . Available at http://www.math.uiuc.edu/Macaulay2/

work page

[15] [15]

Enrico Fermi

M. Hein, W. D¨ ur, E. J., R. R., M. Van den Nest, and B. H.-J., Quantum computers, algorithms and chaos, Proceedings of the International School of Physics “Enrico Fermi”, vol. 162, IOS Press, 2006

work page 2006

[16] [16]

Helwig, Absolutely maximally entangled qudit graph states , 2013

W. Helwig, Absolutely maximally entangled qudit graph states , 2013

work page 2013

[17] [17]

, Absolutely maximally entangled qudit graph states , 2013

work page 2013

[18] [18]

Helwig, W

W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.-K. Lo, Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A 86 (November 2012), 052335

work page 2012

[19] [19]

Higuchi and A

A. Higuchi and A. Sudbery, How entangled can two couples get? , Physics Letters A 273 (2000), no. 4, 213–217

work page 2000

[20] [20]

Huber and N

F. Huber and N. Wyderka, Table of AME states . Available at http://www.tp.nt.uni-siegen.de/ +fhuber/ame.html

work page

[21] [21]

Huber, C

F. Huber, C. Eltschka, J. Siewert, and O. G¨ uhne,Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity , Journal of Physics A: Mathematical and Theoretical 51 (2018), no. 17, 175301

work page 2018

[22] [22]

Huber and M

F. Huber and M. Grassl, Quantum codes of maximal distance and highly entangled subspaces , Quantum 4 (202006), 284

work page

[23] [23]

Huber, O

F. Huber, O. G¨ uhne, and J. Siewert,Absolutely maximally entangled states of seven qubits do not exist , Phys. Rev. Lett. 118 (2017), 200502

work page 2017

[24] [24]

Luque and J.-Y

J.-G. Luque and J.-Y. Thibon, Algebraic invariants of five qubits , Journal of Physics A: Mathematical and General 39 (2005), no. 2, 371

work page 2005

[25] [25]

Martin, O

J. Martin, O. Giraud, P. A. Braun, D. Braun, and T. Bastin, Multiqubit symmetric states with high geometric entanglement, Phys. Rev. A 81 (2010), 062347

work page 2010

[26] [26]

Oeding and S

L. Oeding and S. Sam, Equations for the fifth secant variety of segre products of projective spaces , Experimental Mathematics 25 (2016), no. 1, 94–99

work page 2016

[27] [27]

Oeding and I

L. Oeding and I. Tan, Four-qubit critical states , Journal of Physics A: Mathematical and Theoretical 58 (2025), no. 26, 265301

work page 2025

[28] [28]

1, 33–57

, Tensor decompositions with applications to local unitary and stochastic local operations and classical communication equivalence of multipartite pure states , SIAM Journal on Applied Algebra and Geometry 9 (2025), no. 1, 33–57

work page 2025

[29] [29]

E. M. Rains, Nonbinary quantum codes , IEEE Transactions on Information Theory 45 (1999), no. 6, 1827–1832

work page 1999

[30] [30]

1, 266–271

, Quantum codes of minimum distance two, IEEE Transactions on Information Theory45 (1999), no. 1, 266–271

work page 1999

[31] [31]

E. M. Rains, Quantum weight enumerators , IEEE Trans. Inf. Theory 44 (1996), 1388–1394

work page 1996

[32] [32]

J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics , 3rd ed., Cambridge University Press, 2020

work page 2020

[33] [33]

C. Spee, J. I. de Vicente, D. Sauerwein, and B. Kraus, Entangled pure state transformations via local operations assisted by finitely many rounds of classical communication, Phys. Rev. Lett. 118 (2017Jan), 040503

work page

[34] [34]

Sturmfels, Algorithms in Invariant Theory , Springer Vienna, 2008

B. Sturmfels, Algorithms in Invariant Theory , Springer Vienna, 2008

work page 2008

[35] [35]

´E B Vinberg, The Weyl group of a graded Lie algebra , Mathematics of the USSR-Izvestiya 10 (1976), no. 3, 463

work page 1976

[36] [36]

N. R. Wallach, Geometric Invariant Theory: Over the Real and Complex Numbers , Springer Cham, 2017

work page 2017

[37] [37]

W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits , Phys. Rev. Lett. 80 (1998), 2245–2248

work page 1998

[38] [38]

Y. Zang, Z. Tian, S.-M. Fei, and H.-J. Zuo, Quantum k-uniform states from quantum orthogonal arrays , International Journal of Theoretical Physics 62 (2023)

work page 2023

[39] [39]

D. ˇZ. ¯Dokovi´ c and A. Osterloh,On polynomial invariants of several qubits , Journal of Mathematical Physics 50 (200903), no. 3, 033509

work page