Three qubits can be entangled in two inequivalent ways

TL;DR

The paper classifies pure multipartite entanglement under stochastic LOCC (SLOCC), focusing on single-copy transformations. It proves that three qubits admit two inequivalent genuinely tripartite SLOCC classes, represented by $|GHZ\rangle$ and $|W\rangle$, distinguished by their minimal product-term decompositions and by the nonzero vs zero 3-tangle $\tau$. Through monotones like the entropy of entanglement, $E_2$, and the 3-tangle, it shows these classes cannot be interconverted via LOCC, and it analyzes the robustness of $|W\rangle$ against tracing out a qubit, showing maximal residual bipartite entanglement. The work then generalizes to $N$ parties, proving that only for $N=3$ can a finite SLOCC classification exist in general, while for $N\ge 4$ the class space becomes infinite; it also constructs the $|W_N\rangle$ states, which retain bipartite entanglement across reduced two-qubit subsystems. Overall, the paper provides a comprehensive framework for understanding how entanglement types differ under local operations and how this structure extends to larger multipartite systems.

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.

Figures (1)

• Figure 1: Different local classes of tripartite pure states. The direction of the arrows indicate which non--invertible transformations between classes are possible.