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

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.

💡 Research Summary

The paper provides a complete classification of two important families of multipartite quantum states: four‑qubit pure quantum error‑correcting codes (which are exactly the 1‑uniform or “critical” states) and five‑qubit absolutely maximally entangled (AME) states (the 2‑uniform states). The authors combine techniques from quantum coding theory, Lie algebra theory (especially Vinberg’s Z₂‑graded construction of so(8)), and classical invariant theory to obtain rigorous, constructive results.

First, they recall that a pure quantum error‑correcting code of parameters ((n,K,d)) is equivalent to a (d‑1)‑uniform subspace, and that an AME state is a pure code with maximal uniformity r=⌊n/2⌋. Using Rains’ quantum Singleton bound they identify the unique maximal‑distance‑separable (MDS) codes for the relevant parameters: ((4,4,2))₂, ((5,2,3))₂, and ((6,1,4))₂. The uniqueness is already known up to local unitaries, but the authors strengthen it by showing that every 4‑qubit pure code is locally equivalent to a subspace of the unique ((4,4,2)) code (denoted C₂), and every 5‑qubit AME state is locally equivalent to a point in the unique ((5,2,3)) code (denoted C₁).

The second major contribution is the description of the equivalence classes inside C₂ and C₁. For C₂ the relevant symmetry group is the Weyl group of type D₄, of order 144, acting on the Cartan subalgebra of the Z₂‑graded Lie algebra so(8) whose grade‑1 piece is (ℂ²)⊗⁴. Two 4‑qubit critical states are equivalent under local SU(2)⁴ iff they are related by this Weyl group. For C₁ the symmetry group is a smaller order‑24 group W = ⟨iX, iZ, Q⟩, where X and Z are Pauli matrices and Q = ½