Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity

TL;DR

This paper establishes new bounds on the existence of absolutely maximally entangled states in higher dimensions using shadow inequalities and introduces a quantum MacWilliams identity in the Bloch representation.

Contribution

It develops bounds on AME states beyond qubits, derives the quantum MacWilliams identity, and provides an example of a mixed-dimension AME state.

Findings

New bounds on AME state existence in higher dimensions

Derivation of the quantum MacWilliams identity in the Bloch representation

An example of a 2×3×3×3 AME state with maximal bipartite entanglement

Abstract

A pure multipartite quantum state is called absolutely maximally entangled (AME), if all reductions obtained by tracing out at least half of its parties are maximally mixed. Maximal entanglement is then present across every bipartition. The existence of such states is in many cases unclear. With the help of the weight enumerator machinery known from quantum error correction and the generalized shadow inequalities, we obtain new bounds on the existence of AME states in dimensions larger than two. To complete the treatment on the weight enumerator machinery, the quantum MacWilliams identity is derived in the Bloch representation. Finally, we consider AME states whose subsystems have different local dimensions, and present an example for a $2 \times3 \times 3 \times 3$ system that shows maximal entanglement across every bipartition.