On two maximally entangled couples

TL;DR

This paper reviews and provides multiple proofs that no four-qubit absolutely maximally entangled (AME) state exists, using invariant theory, coding theory, and linear algebra.

Contribution

It offers new and old proofs of the non-existence of four-qubit AME states, integrating diverse mathematical approaches.

Findings

No four-qubit AME state exists.

Multiple proof techniques confirm the non-existence.

The proofs utilize invariant theory, coding theory, and linear algebra.

Abstract

In a seminal article, Higuchi and Sudbery showed that a pure four-qubit state can not be maximally entangled across every bipartition. Such states are now known as absolutely maximally entangled (AME) states. Here we give a series of old and new proofs of the fact that no four-qubit AME state exists. These are based on invariant theory, methods from coding theory, and basic properties from linear algebra such as the Pauli commutation relations.