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

Ian Tan

Published Jun 26, 2026• 0 citations
Publisher

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 $$\mathcal {C}$$ .

DOI
10.1007/s44464-026-00021-z

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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 .
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 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. 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⟩|Φ\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⟩|Φ\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⟩|Φ\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⟩|Φ\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.
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.
As emerging quantum architectures evolve into heterogeneous networks combining different physical substrates, such as qubits for logic and higher-dimensional qudits for robust communication, the traditional scalar metrics of quantum error correction become insufficient. To address this, we introduce a mathematical framework based on dimension multisets to characterize quantum error-correcting codes (QECC) and absolutely maximally entangled (AME) states in mixed-dimensional Hilbert spaces. By replacing scalar weights with multisets, we accurately capture the exact physical composition of error supports across these diverse systems. Our central result is the mixed-dimensional quantum MacWilliams identity, which establishes the formal algebraic relationship between Shor-Laflamme enumerators and unitary weight enumerators. From this foundation, we deduce the mixed-dimensional shadow identity and derive rigorous, generalized constraints on code parameters, explicitly formulating the mixed-dimensional quantum Hamming, Singleton and Scott bounds, and developing a linear program to systematically evaluate code viability. For the Singleton bound, a tighter bound that has no homogeneous analogue is derived for pure mixed-dimensional codes. Finally, we deploy this enumerator machinery to thoroughly analyze AME states, utilizing shadow inequalities to constrain their existence and introducing a combinatorial grid method for the explicit construction of mixed-dimensional tripartite AME states.
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.