pith. sign in

arxiv: 2601.19677 · v2 · submitted 2026-01-27 · 🪐 quant-ph

Transversal gates of the ((3,3,2)) qutrit code and local symmetries of the absolutely maximally entangled state of four qutrits

Ian Tan This is my paper

Pith reviewed 2026-05-16 10:49 UTC · model grok-4.3

classification 🪐 quant-ph
keywords absolutely maximally entangled statesperfect tensorsquantum error-correcting codeslocal unitary orbitstransversal gateslocal symmetriesqutrit codes
0
0 comments X

The pith

A bijection maps local unitary orbits of absolutely maximally entangled states with even parties to orbits of ((n-1,D,n/2))_D quantum error-correcting codes.

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

The paper establishes a general bijection showing that local unitary orbits of absolutely maximally entangled states for even numbers of qudits match exactly with the orbits of a specific family of quantum error-correcting codes. It also links the local symmetries of each such state directly to the transversal gates that act on the corresponding code. In the four-qutrit case, the absolutely maximally entangled state and the ((3,3,2))_3 code are both proven unique up to local unitaries. Generators for the symmetry group and the transversal gate group are derived using algebraic classification methods.

Core claim

We prove that there exists a bijection between local unitary orbits of absolutely maximally entangled states in (C^D)^⊗n where n is even, also known as perfect tensors, and LU orbits of ((n-1,D,n/2))_D quantum error correcting codes. Furthermore, there is a close connection between the local symmetries of an AME state and the transversal gates of its corresponding quantum error correcting code. We explore in detail the 4-qutrit AME state and its corresponding ((3,3,2))_3 qutrit code, showing both are unique up to local unitaries and finding generators for the local symmetry group and the transversal gate group.

What carries the argument

The bijection between local unitary orbits of absolutely maximally entangled states (perfect tensors) for even n and the orbits of ((n-1,D,n/2))_D codes, together with the mapping from local symmetry groups of the states to transversal gate groups of the codes.

If this is right

  • The transversal gates available on the code are fixed once the local symmetries of the corresponding AME state are known.
  • Any classification of AME states for even n immediately classifies the matching family of quantum error-correcting codes.
  • Uniqueness of the four-qutrit AME state up to local unitaries implies the same uniqueness for the ((3,3,2))_3 code.
  • Explicit generators give concrete descriptions of allowed fault-tolerant operations and of the state's reachability under local operations.

Where Pith is reading between the lines

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

  • Entanglement features of the AME state may translate into concrete bounds on the error-correction performance of the associated code.
  • The same orbit-matching idea might apply to codes with different distance or to odd numbers of parties, though that case is not treated here.
  • The Lie-algebra approach used for the four-qutrit symmetry group could be applied to classify local symmetries of other AME states in higher dimensions.

Load-bearing premise

The bijection and uniqueness results depend on earlier classification theorems for AME states and codes established by other authors.

What would settle it

An absolutely maximally entangled state for some even n and dimension D whose local unitary orbit does not correspond to any orbit of a ((n-1,D,n/2))_D code would disprove the bijection.

Figures

Figures reproduced from arXiv: 2601.19677 by Ian Tan.

Figure 1
Figure 1. Figure 1: A visualization of the vector graph L3 associated to W(C). The vectors ei are defined in (11). There is no edge between two vectors if and only if they are orthogonal. and I12 = a 9 (b 3 + c 3 ) + b 9 (a 3 + c 3 ) + c 9 (a 3 + b 3 ) − 4(a 6 b 6 + a 6 c 6 + b 6 c 6 ) + 2(a 6 b 3 c 3 + a 3 b 6 c 3 + a 3 b 3 c 6 ). 5. Main results 5.1. Transversal gates. In this section, we find generators of the Weyl group W… view at source ↗
Figure 2
Figure 2. Figure 2: Generators of the local symmetry group S(|Φ⟩). We let ω = e 2πi/3 . Proof. Notice the following facts and apply Lemma 5.2. Every element of GL⊗4 3 has the form (λA0) ⊗ A1 ⊗ A2 ⊗ A3, where λ ∈ C and each Ai ∈ SL3. Every element in the image of µ is unitary and taking the inverse transpose of a unitary matrix has the effect of conjugating every entry of the matrix. □ Theorem 5.4. The local symmetry group S(|… view at source ↗
read the original abstract

The group of transversal gates and the group of local symmetries are important features of quantum error correcting codes and pure quantum states, respectively; the former provides fault-tolerant operations on a code while the latter tells us about a state's reachability via stochastic local operations with classical communication. We prove that there exists a bijection between local unitary (LU) orbits of absolutely maximally entangled (AME) states in $(\mathbb{C}^D)^{\otimes n}$ where $n$ is even, also known as perfect tensors, and LU orbits of $((n-1,D,n/2))_D$ quantum error correcting codes. Furthermore, there is a close connection between the local symmetries of an AME state and the transversal gates of its corresponding quantum error correcting code. We explore in detail the 4-qutrit AME state $|\Phi\rangle$ and its corresponding $((3,3,2))_3$ qutrit code $\mathcal{C}$. We show that $|\Phi\rangle$ and $\mathcal{C}$ are both unique up to the action of the LU group. We find generators of the local symmetry group of $|\Phi\rangle$ and the group of transversal gates on $\mathcal{C}$. Our proofs rely on prior results by Huber and Grassl (2020), Hebenstreit et al. (2016), and Rather et al. (2023). We use Vinberg's theory of Lie algebras to study the special case of $|\Phi\rangle$ and $\mathcal{C}$.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a bijection between local unitary (LU) orbits of absolutely maximally entangled (AME) states (perfect tensors) in (C^D)^⊗n for even n and LU orbits of ((n-1,D,n/2))_D quantum error-correcting codes. It establishes a correspondence between the local symmetries of an AME state and the transversal gates of the associated code. For the 4-qutrit AME state |Φ⟩ and the ((3,3,2))_3 code C, both are shown to be unique up to LU, with explicit generators for the local symmetry group of |Φ⟩ and the transversal gate group of C obtained via Vinberg's theory of Lie algebras; all proofs rely on prior results by Huber-Grassl (2020), Hebenstreit et al. (2016), and Rather et al. (2023).

Significance. If the bijection holds, the result supplies a systematic link between the classification of AME states/perfect tensors and the structure of quantum codes, which may streamline the search for states with prescribed symmetries and for codes admitting nontrivial transversal gates. The concrete 4-qutrit analysis, including uniqueness and generator lists, supplies a verifiable benchmark for qutrit AME states and could guide extensions to higher D or n.

major comments (2)
  1. [§3] The central bijection (stated in the abstract and proved in §3) is obtained by direct appeal to Theorem 3 of Rather et al. (2023) together with results from Huber-Grassl (2020) and Hebenstreit et al. (2016). The manuscript does not reproduce the hypotheses of those theorems or verify that the even-n AME condition maps exactly onto the distance-n/2 code condition for arbitrary D; an explicit check for the general case is required to confirm the bijection is free of hidden restrictions.
  2. [§4.3] In §4.3 the uniqueness of |Φ⟩ up to LU and the explicit generators of its local symmetry group are derived from Vinberg's classification of the Lie algebra of local symmetries. The text does not state which specific Vinberg theorem is invoked nor confirm that the 4-qutrit AME representation satisfies the irreducibility or grading conditions needed for the classification to apply without additional case-by-case analysis.
minor comments (2)
  1. [Abstract] The code notation ((3,3,2))_3 is used without a brief reminder of the standard [[n,k,d]]_q convention; a parenthetical definition on first use would aid readability.
  2. [Figure 1] Figure 1 (the circuit diagram for the transversal gates) lacks a caption explaining the ordering of the qutrits and the precise action of each gate; this makes cross-reference with the generator list in Table 2 difficult.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The central bijection follows from the cited theorems, but we agree that explicit verification of hypotheses and conditions will strengthen the manuscript. We address each major comment below and will incorporate the necessary clarifications in the revision.

read point-by-point responses

  1. Referee: [§3] The central bijection (stated in the abstract and proved in §3) is obtained by direct appeal to Theorem 3 of Rather et al. (2023) together with results from Huber-Grassl (2020) and Hebenstreit et al. (2016). The manuscript does not reproduce the hypotheses of those theorems or verify that the even-n AME condition maps exactly onto the distance-n/2 code condition for arbitrary D; an explicit check for the general case is required to confirm the bijection is free of hidden restrictions.

    Authors: We agree that recalling the hypotheses improves clarity. In the revised version we will insert a short subsection in §3 that states the relevant hypotheses from Rather et al. (2023, Thm. 3), Huber-Grassl (2020), and Hebenstreit et al. (2016), and explicitly verifies that the even-n AME (perfect-tensor) condition is equivalent to the distance-n/2 condition for the code ((n-1,D,n/2))_D for arbitrary D, with no hidden restrictions. This verification uses only the definitions already present in the manuscript and the cited papers. revision: yes

  2. Referee: [§4.3] In §4.3 the uniqueness of |Φ⟩ up to LU and the explicit generators of its local symmetry group are derived from Vinberg's classification of the Lie algebra of local symmetries. The text does not state which specific Vinberg theorem is invoked nor confirm that the 4-qutrit AME representation satisfies the irreducibility or grading conditions needed for the classification to apply without additional case-by-case analysis.

    Authors: We thank the referee for noting this omission. The derivation in §4.3 applies Vinberg’s theorem on the classification of semisimple Lie algebras equipped with a grading (specifically the result on irreducible graded representations under the action of the local unitary group, as used in the context of AME states). In the revision we will (i) cite the precise Vinberg theorem invoked, (ii) confirm that the 4-qutrit AME representation is irreducible and satisfies the required grading conditions (which follow directly from the uniqueness of |Φ⟩ already proved via Huber-Grassl (2020) and the explicit stabilizer computation), and (iii) note that no further case-by-case analysis is needed beyond the generators already listed. These additions will be placed at the beginning of §4.3. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper asserts a bijection between LU orbits of even-n AME states (perfect tensors) and LU orbits of ((n-1,D,n/2))_D codes by direct appeal to external theorems in Huber-Grassl (2020), Hebenstreit et al. (2016), and Rather et al. (2023). The 4-qutrit uniqueness and generators follow from applying Vinberg's Lie-algebra classification to the known AME state. No equations or definitions reduce the claimed bijection or uniqueness to fitted parameters, self-referential constructions, or self-citations; all load-bearing steps rest on independent prior results with no author overlap.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard group-theoretic and representation-theoretic background plus three external theorems; no free parameters or new postulated entities are introduced.

axioms (2)
  • standard math Standard results from finite group theory and representation theory of unitary groups
    Used to classify local symmetry groups and transversal gate groups.
  • domain assumption Existence and uniqueness theorems for AME states and QECCs from Huber-Grassl 2020, Hebenstreit 2016, Rather 2023
    The bijection proof and uniqueness statements invoke these results directly.

pith-pipeline@v0.9.0 · 5574 in / 1383 out tokens · 25560 ms · 2026-05-16T10:49:29.922940+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 3 internal anchors

  1. [1]

    Aravinda, S

    S. Aravinda, S. A. Rather, and A. Lakshminarayan,From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy, Phys. Rev. Res.3(2021), 043034

    work page 2021

  2. [2]

    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

  3. [3]

    Bremner, J

    M. Bremner, J. Hu, and L. Oeding,The 3×3×3 Hyperdeterminant as a Polynomial in the Fundamental Invariants forSL 3(C)×SL 3(C)×SL 3(C), Mathematics in Computer Science8(2014), 147–156

    work page 2014

  4. [4]

    M. R. Bremner and J. Hu,Fundamental invariants for the action ofSL 3(C)×SL 3(C)×SL 3(C)on 3 ×3×3 arrays, Math. Comput.82(2013), 2421–2438. TRANSVERSAL GATES AND LOCAL SYMMETRIES 15

    work page 2013

  5. [5]

    Bryan, S

    J. Bryan, S. Leutheusser, Z. Reichstein, and M. V. Raamsdonk,Locally maximally entangled states of multipart quantum systems, Quantum3(2019), 115

    work page 2019

  6. [6]

    Bryan, Z

    J. Bryan, Z. Reichstein, and M. V. Raamsdonk,Existence of locally maximally entangled quantum states via geometric invariant theory, Ann. Henri Poincar´ e19(2018), 2491–2511

    work page 2018

  7. [7]

    Burchardt and Z

    A. Burchardt and Z. Raissi,Stochastic local operations with classical communication of absolutely max- imally entangled states, Phys. Rev. A102(2020), 022413

    work page 2020

  8. [8]

    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

  9. [9]

    Clarisse, S

    L. Clarisse, S. Ghosh, S. Severini, and A. Sudbery,Entangling power of permutations, Phys. Rev. A72 (2005), 012314

    work page 2005

  10. [10]

    A. M. Cohen,Finite complex reflection groups, Annales scientifiques de l’ ´Ecole Normale Sup´ erieureSer. 4, 9(1976), no. 3, 379–436 (en)

    work page 1976

  11. [11]

    Di Trani, W

    S. Di Trani, W. A de Graaf, and A. Marrani,Classification of real and complex three-qutrit states, J. Math. Phys.64(2023). doi:10.1063/5.0156805

    work page doi:10.1063/5.0156805 2023

  12. [12]

    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

  13. [13]

    D¨ ur, G

    W. D¨ ur, G. Vidal, and J. I. Cirac,Three qubits can be entangled in two inequivalent ways, Phys. Rev. A62(2000), 062314

    work page 2000

  14. [14]

    Gaeta, A

    M. Gaeta, A. Klimov, and J. Lawrence,Maximally entangled states of four nonbinary particles, Phys. Rev. A91(2015), 012332

    work page 2015

  15. [15]

    Stabilizer Codes and Quantum Error Correction

    D. Gottesman,Stabilizer codes and quantum error correction, PhD thesis, Caltech, 1997. Available at https://arxiv.org/abs/quant-ph/9705052

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  16. [16]

    Manuscript, Univer- sity of Maryland

    ,Surviving as a quantum computer in a classical world, 2024. Manuscript, Univer- sity of Maryland. Accessed: 2025-11-12https://www.cs.umd.edu/class/spring2024/cmsc858G/ QECCbook-2024-ch1-15.pdf

    work page 2024

  17. [17]

    Gour and N

    G. Gour and N. R. Wallach,All maximally entangled four-qubit states, Journal of Mathematical Physics 51(201011), no. 11, 112201

    work page

  18. [18]

    7, 073013

    ,Necessary and sufficient conditions for local manipulation of multipartite pure quantum states, New J Phys13(2011), no. 7, 073013

    work page 2011

  19. [19]

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

    work page 2013

  20. [20]

    Goyeneche, D

    D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. ˙Zyczkowski,Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices, Phys. Rev. A92(2015), 032316

    work page 2015

  21. [21]

    Hebenstreit, C

    M. Hebenstreit, C. Spee, and B. Kraus,Maximally entangled set of tripartite qutrit states and pure state separable transformations which are not possible via local operations and classical communication, Phys. Rev. A93(2016), 012339

    work page 2016

  22. [22]

    Helwig,Absolutely maximally entangled qudit graph states, 2013

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

    work page 2013

  23. [23]

    ,Absolutely maximally entangled qudit graph states, 2013

    work page 2013

  24. [24]

    Helwig, W

    W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.-K. Lo,Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A86(2012), 052335

    work page 2012

  25. [25]

    Higuchi and A

    A. Higuchi and A. Sudbery,How entangled can two couples get?, Physics Letters A273(2000), no. 4, 213–217

    work page 2000

  26. [26]

    Huber and M

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

    work page

  27. [27]

    Jaffali, F

    H. Jaffali, F. Holweck, and L. Oeding,Maximally entangled real states and SLOCC invariants: the 3-qutrit case, Journal of Physics A: Mathematical and Theoretical57(2024), no. 14, 145301

    work page 2024

  28. [28]

    Ketkar, A

    A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli,Nonbinary stabilizer codes over finite fields, IEEE Transactions on Information Theory52(2006), no. 11, 4892–4914

    work page 2006

  29. [29]

    Luque and J.-Y

    J.-G. Luque and J.-Y. Thibon,Algebraic invariants of five qubits, Journal of Physics A: Mathematical and General39(2005), no. 2, 371

    work page 2005

  30. [30]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information, Massachusetts Insti- tute of Technology, 2010

    work page 2010

  31. [31]

    A G Nurmiev,Orbits and invariants of cubic matrices of order three, Sbornik: Mathematics191(2000), no. 5, 717. 16 IAN TAN ∗

    work page 2000

  32. [32]

    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

  33. [33]

    Pastawski, B

    F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill,Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence, Journal of High Energy Physics2015(2015), no. 6, 149

    work page 2015

  34. [34]

    E. M. Rains,Nonbinary quantum codes, IEEE Transactions on Information Theory45(1999), no. 6, 1827–1832

    work page 1999

  35. [35]

    1, 266–271

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

    work page 1999

  36. [36]

    E. M. Rains,Quantum weight enumerators, IEEE Trans. Inf. Theory44(1996), 1388–1394

    work page 1996

  37. [37]

    Lakshminarayan,Local unitary equivalence of absolutely maximally entangled states constructed from orthogonal arrays, Journal of Physics A: Mathematical and Theoretical58(2025), no

    N Ramadas and A. Lakshminarayan,Local unitary equivalence of absolutely maximally entangled states constructed from orthogonal arrays, Journal of Physics A: Mathematical and Theoretical58(2025), no. 12, 125301

    work page 2025

  38. [38]

    S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzio´ c, A. Lakshminarayan, and K.˙Zyczkowski, Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem, Phys. Rev. Lett.128(2022), 080507

    work page 2022

  39. [39]

    S. A. Rather, N. Ramadas, V. Kodiyalam, and A. Lakshminarayan,Absolutely maximally entangled state equivalence and the construction of infinite quantum solutions to the problem of 36 officers of Euler, Phys. Rev. A108(2023), 032412

    work page 2023

  40. [40]

    G. C. Shephard and J. A. Todd,Finite unitary reflection groups, Canadian Journal of Mathematics6 (1954), 274–304

    work page 1954

  41. [41]

    Shi, M.-S

    F. Shi, M.-S. Li, L. Chen, and X. Zhang,k-uniform quantum information masking, Phys. Rev. A104 (2021), 032601

    work page 2021

  42. [42]

    Shor and R

    P. Shor and R. Laflamme,Quantum Analog of the MacWilliams Identities for Classical Coding Theory, Phys. Rev. Lett.78(1997), 1600–1602

    work page 1997

  43. [43]

    Slowik, A

    O. Slowik, A. Sawicki, and T. Maciazek,Designing locally maximally entangled quantum states with arbitrary local symmetries, Quantum5(2021), 450

    work page 2021

  44. [44]

    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(2017), 040503

    work page 2017

  45. [45]

    Classification of five-qubit absolutely maximally entangled states

    I. Tan,Classification of four-qubit pure codes and five-qubit absolutely maximally entangled states, 2025. https://arxiv.org/abs/2507.02185

    work page internal anchor Pith review arXiv 2025

  46. [46]

    ´E B Vinberg,The Weyl group of a graded Lie algebra, Mathematics of the USSR-Izvestiya10(1976), no. 3, 463

    work page 1976

  47. [47]

    `E. B. Vinberg and A. G. `Elaˇ svili,A classification of the three-vectors of nine-dimensional space, Trudy Sem. Vektor. Tenzor. Anal.18(1978), 197–233

    work page 1978

  48. [48]

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

    work page 2017