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arXiv:2601.19677

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

Ian Tan

quant-phSubmitted Jan 27, 2026• Announced Jan 28, 2026• 0 citations
v1 · Jan 27, 2026v2 · May 6, 2026
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Abstract

We provide a proof 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. Thus, by a result of Rather et al. (2023), the AME state of 4 qutrits and the pure $((3,3,2))_3$ qutrit code $\mathcal{C}$ are both unique up to the action of the LU group. We further explore the connection between the 4-qutrit AME state and the code $\mathcal{C}$ by showing that the group of transversal gates of $\mathcal{C}$ and the group of local symmetries of the AME state are closely related. Taking advantage of results from Vinberg's theory of graded Lie algebras, we find generators of both of these groups.

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16 pages, 2 figures

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We provide a proof that there exists a bijection between local unitary (LU) orbits of absolutely maximally entangled (AME) states in (CD)⊗n(\mathbb{C}^D)^{\otimes n}(CD)⊗n where nnn is even, also known as perfect tensors, and LU orbits of ((n−1,D,n/2))D((n-1,D,n/2))_D((n−1,D,n/2))D​ quantum error correcting codes. Thus, by a result of Rather et al. (2023), the AME state of 4 qutrits and the pure ((3,3,2))3((3,3,2))_3((3,3,2))3​ qutrit code C\mathcal{C}C are both unique up to the action of the LU group. We further explore the connection between the 4-qutrit AME state and the code C\mathcal{C}C by showing that the group of transversal gates of C\mathcal{C}C and the group of local symmetries of the AME state are closely related. Taking advantage of results from Vinberg's theory of graded Lie algebras, we find generators of both of these groups.
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 (CD)⊗n(\mathbb {C}^D)^{\otimes n}(CD)⊗n where n is even, also known as perfect tensors, and LU orbits of ((n−1,D,n/2))D((n-1,D,n/2))_D((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 ∣Φ⟩|\Phi \rangle ∣Φ⟩ and its corresponding ((3,3,2))3((3,3,2))_3((3,3,2))3​ qutrit code C\mathcal {C}C . We show that ∣Φ⟩|\Phi \rangle ∣Φ⟩ and C\mathcal {C}C are both unique up to the action of the LU group. We find generators of the local symmetry group of ∣Φ⟩|\Phi \rangle ∣Φ⟩ and the group of transversal gates on C\mathcal {C}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 ∣Φ⟩|\Phi \rangle ∣Φ⟩ and C\mathcal {C}C .
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 C\mathcal{C}C, and that two such states are equivalent if and only if they are related by the action of a transversal gate of C\mathcal{C}C. Furthermore, we exhibit a set of three invariant polynomials that separates these equivalence classes. As auxiliary results, we construct a 3-uniform nnn-qubit state for even n≥6n\geq 6n≥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 8×88\times 88×8 skew-symmetric matrices, allowing us to apply results from Vinberg's theory of graded Lie algebras.
Dual unitary gates are highly non-local two-qudit unitary gates that have been studied extensively in quantum many-body physics and quantum information in the recent past. A special class of dual unitary gates consists of rank-four perfect tensors that are equivalent to highly entangled multipartite pure states called absolutely maximally entangled (AME) states. In this work, numerical and analytical constructions of dual unitary gates and perfect tensors that are diagonal in a special maximally entangled basis are presented. The main ingredient in our construction is a phase-valued (unimodular) two-dimensional array whose discrete Fourier transform is also unimodular. We obtain perfect tensors for several local Hilbert space dimensions, particularly, in dimension six. A perfect tensor in local dimension six is equivalent to an AME state of four qudits, denoted as AME(4,6). Such a state cannot be constructed from existing constructions of AME states based on error-correcting codes and graph states. An explicit construction of AME(4,6) states is provided in this work using two-qudit controlled and single-qudit gates making it feasible to generate such states experimentally.
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.
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.
Ordering and classifying multipartite quantum states by their entanglement content remains an open problem. One class of highly entangled states, useful in quantum information protocols, the absolutely maximally entangled (AME) ones, are especially hard to compare as all their subsystems are maximally random. While it is well known that there is no AME state of four qubits, many analytical examples and numerically generated ensembles of four-qutrit AME states are known. However, we prove the surprising result that there is truly only one AME state of four qutrits up to local unitary equivalence. In contrast, for larger local dimensions, the number of local unitary classes of AME states is shown to be infinite. Of special interest is the case of local dimension 6, where it was established recently that a four-party AME state does exist, providing a quantum solution to the classically impossible Euler problem of 36 officers. Based on this, an infinity of quantum solutions are constructed, and we prove that these are not equivalent. The methods developed can be usefully generalized to multipartite states of any number of particles.