$n$ qubits can be entangled in two different ways

TL;DR

This paper introduces a new classification method for pure states of n qubits based on basis state matrices, providing a clear criterion to distinguish between different types of genuine entanglement.

Contribution

It proposes a canonical form of the basis state matrix and a necessary and sufficient condition for genuine entanglement classification of n-qubit states.

Findings

Genuinely entangled states are partitioned into two families.

States with BSM not transformable into the canonical form are always genuinely entangled.

GHZ and W states belong to the first family, cluster states to the second.

Abstract

In [M. Walter et al., Science 340, 1205, 7 June (2013)], via polytopes they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification. In this paper, we study entanglement classification of pure states of $n$ qubits via the basis state matrix (BSM) whose rows are the basis states. We propose a canonical form of BSM obtained by exchanging columns (i.e. permutation of qubits) and rows of BSM and then a necessary and sufficient condition for a genuinely entangled state of n qubits via a canonical form of BSM. Thus, for any $n$ qubits, the genuinely entangled states can be partitioned into two families. One family includes all states whose BSM cannot be transformed into the canonical form. The states with the BSM are always genuinely entangled no matter what the non-zero coefficients are. GHZ and W states belong to the family. The other includes all states whose BSM can be transformed into the canonical form, but for any canonical form of BSM, some two columns or rows of the corresponding coefficient matrix are not proportional. The cluster state belongs to the family.