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

• URL: http://arxiv.org/abs/2601.19677v1
• Date: Tue, 27 Jan 2026 14:58:07 GMT
• Title: Transversal gates of the ((3,3,2)) qutrit code and local symmetries of the absolutely maximally entangled state of four qutrits
• Authors: Ian Tan,
• Abstract summary: Local unitary orbits of absolutely maximally entangled (AME) states in $(mathbbCD)otimes n where $n$ is even, also known as perfect tensors, are studied.<n>We show that the group of gates of $mathcalC$ and the group of local symmetries of the AME state are closely related.
• Score: 0.30458514384586394
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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