Multipartite Entanglement and Hyperdeterminants

TL;DR

This paper introduces a unified classification of multipartite entanglement using hyperdeterminants, revealing a complex structure of entangled states and challenging the notion of canonical maximally entangled states in systems with four or more qubits.

Contribution

It provides a novel framework for classifying multipartite entanglement via hyperdeterminants, highlighting the partial order of entangled classes and their relation to Bell inequalities.

Findings

Multipartite entanglement classes form a partial order under local actions.

The generic entangled class with nonzero hyperdeterminant does not include maximally entangled states in Bell's inequalities.

Maximally entangled states are not always locally interconvertible with other entangled states in systems with four or more qubits.

Abstract

We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the 4 or more qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for that of mixed states.