Classification of four-qubit pure codes and five-qubit absolutely maximally entangled states

• URL: http://arxiv.org/abs/2507.02185v2
• Date: Sat, 12 Jul 2025 04:28:52 GMT
• Title: Classification of four-qubit pure codes and five-qubit absolutely maximally entangled states
• Abstract summary: We prove that every 5-qubit absolutely maximally entangled (AME) states is equivalent by a local unitary transformation to a point in the unique ((5,2,3)) quantum error code C.<n>We also show that every 4-qubit pure code is equivalent to a subspace of the unique ((4,2)) and construct an infinite family of 3-uniform n-qubit states for even $ngeq 6$.
• Score: 0.6526824510982802
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove that every 5-qubit absolutely maximally entangled (AME) state is equivalent by a local unitary transformation to a point in the unique ((5,2,3)) quantum error correcting code C. Furthermore, two points in C are equivalent if and only if they are related by a group of order 24 acting on C. There exists a set of 3 invariant polynomials that separates equivalence classes of 5-qubit AME states. We also show that every 4-qubit pure code is equivalent to a subspace of the unique ((4,4,2)) and construct an infinite family of 3-uniform n-qubit states for even $n\geq 6$. The proofs rely heavily on results from Vinberg and classical invariant theory.

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