The inductive entanglement classification yields ten rather than eight classes of four-qubit entangled states

TL;DR

This paper revises the inductive classification of four-qubit entangled states, showing that two previously discarded classes are non-empty, thus confirming a total of ten genuine entanglement classes.

Contribution

It demonstrates that two of the three discarded superclasses are non-empty and provides explicit canonical states, refining the classification scheme.

Findings

Two superclasses previously considered empty are non-empty.

Explicit canonical states are provided for the new classes.

Total of ten genuine four-qubit entanglement classes confirmed.

Abstract

Lamata et al. use an inductive approach to classify the entangled pure states of four qubits under stochastic local operations and classical communication (SLOCC) [PRA 75(2), 022318 (2007)]. The inductive method yields a priori ten different entanglement superclasses, of which they discard three as empty. One of the remaining superclasses is split in two, resulting in eight superclasses of genuine four-qubit entanglement. Here, we show that two of the three discarded superclasses are in fact non-empty and should have been retained. We give explicit expressions for the canonical states for those superclasses, up to SLOCC and qubit permutations. Furthermore, we confirm that the third discarded superclass is indeed empty, yielding a total of ten superclasses of genuine four-qubit entanglement under the inductive classification scheme.