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

Ian Tan

TL;DR

The work addresses the classification of highly entangled quantum codes by marrying quantum information with Vinberg invariant theory. It shows that every 5-qubit AME state is LU-equivalent to a point in the unique ((5,2,3)) code C1, with equivalence governed by a finite group of order 24, and that every 4-qubit pure code is LU-equivalent to a subspace of the unique ((4,4,2)) code C2. The authors develop a Vinberg-theoretic framework to connect 4-qubit state spaces to graded Lie algebras, enabling a precise embedding and the construction of a 3-uniform infinite family of even-n states. They further provide a separating invariant set of polynomials (degrees 6, 8, 12) that fully distinguishes SU(2)⊗5 orbits of 5-qubit AME states, offering a practical criterion for equivalence. Overall, the paper advances the classification of maximally entangled states and their associated quantum codes through a synthesis of invariant theory, Lie theory, and quantum information.

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.

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

TL;DR

The work addresses the classification of highly entangled quantum codes by marrying quantum information with Vinberg invariant theory. It shows that every 5-qubit AME state is LU-equivalent to a point in the unique ((5,2,3)) code C1, with equivalence governed by a finite group of order 24, and that every 4-qubit pure code is LU-equivalent to a subspace of the unique ((4,4,2)) code C2. The authors develop a Vinberg-theoretic framework to connect 4-qubit state spaces to graded Lie algebras, enabling a precise embedding and the construction of a 3-uniform infinite family of even-n states. They further provide a separating invariant set of polynomials (degrees 6, 8, 12) that fully distinguishes SU(2)⊗5 orbits of 5-qubit AME states, offering a practical criterion for equivalence. Overall, the paper advances the classification of maximally entangled states and their associated quantum codes through a synthesis of invariant theory, Lie theory, and quantum information.

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 . The proofs rely heavily on results from Vinberg and classical invariant theory.

Paper Structure

This paper contains 26 sections, 36 theorems, 60 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The main MDS family
  3. Overview
  4. Notation and conventions
  5. Preliminaries
  6. Quantum error correcting codes
  7. Critical states
  8. Rains's construction
  9. Local symmetries of the 6-qubit AME state
  10. A unitary change of coordinates
  11. Construction of 3-uniform states
  12. Stabilizers and fault-tolerant operations
  13. Group of order 24
  14. Lie theory and Vinberg theory
  15. A graded Lie algebra
  16. ...and 11 more sections

Key Result

theorem 1.1

The set of 1-uniform states in $(\mathbb C^2)^{\otimes 4}$ is equal to $\operatorname{SU}_2^{\otimes 4}\mathcal{C}_2$. Two points in $\mathcal{C}_2$ are $\operatorname{SU}_2^{\times 4}$-equivalent if and only if they are $W$-equivalent, where $W$ is the Weyl group of the Lie algebra $\mathfrak{so}_8

Theorems & Definitions (72)

  • theorem 1.1: Classification of 4-qubit critical states
  • proof
  • theorem 1.2: Main theorem
  • Definition 2.1
  • theorem 2.2: Quantum Singleton bound
  • proof
  • theorem 2.3: Rains
  • proof
  • theorem 2.4
  • proof
  • ...and 62 more