Three qubits can be entangled in two inequivalent ways

TLDR

Invertible local transformations define equivalence classes of entangled states, focusing on a single copy’s entanglement properties. Two states share the same entanglement if each can be obtained from the other by LOCC with nonzero probability, and this criterion extends to higher‑dimensional and multipartite systems. Applying this classification to pure three‑qubit states reveals two inequivalent genuine tripartite entanglement classes—GHZ and W—where the W state preserves maximal bipartite entanglement upon tracing out any qubit, and the same result holds for higher‑dimensional and multipartite systems, showing that typical random pure states are not interconvertible by LOCC.

Abstract

Invertible local transformations of a multipartite system are used to define equivalence classes in the set of entangled states. This classification concerns the entanglement properties of a single copy of the state. Accordingly, we say that two states have the same kind of entanglement if both of them can be obtained from the other by means of local operations and classical communcication (LOCC) with nonzero probability. When applied to pure states of a three-qubit system, this approach reveals the existence of two inequivalent kinds of genuine tripartite entanglement, for which the GHZ state and a W state appear as remarkable representatives. In particular, we show that the W state retains maximally bipartite entanglement when any one of the three qubits is traced out. We generalize our results both to the case of higher dimensional subsystems and also to more than three subsystems, for all of which we show that, typically, two randomly chosen pure states cannot be converted into each other by means of LOCC, not even with a small probability of success.