Classification of five-qubit absolutely maximally entangled states
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916026407452672 |
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| author | Tan, Ian |
| author_facet | Tan, Ian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02185 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of five-qubit absolutely maximally entangled states Tan, Ian Quantum Physics Representation Theory 81P70, 15A72 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. |
| title | Classification of five-qubit absolutely maximally entangled states |
| topic | Quantum Physics Representation Theory 81P70, 15A72 |
| url | https://arxiv.org/abs/2507.02185 |