Multipartite Entanglement

Every entangled state you've met so far has had exactly 2 qubits. With 3 or more, "is it entangled?" stops being a yes/no question — there are several genuinely different ways a multi-qubit state can be entangled, or only partly entangled, or not entangled at all. This chapter is the vocabulary for telling them apart, before Ch. 17 dives into the two most famous examples.

Three categories, one test

For an n-qubit state, check every way of splitting the qubits into two groups (a bipartition) and ask, using Ch. 10's partial trace: is each group's reduced state pure (unentangled across that particular split) or mixed (entangled across it)?

• Fully separable: every bipartition is unentangled. The whole state is really just a product of independent pieces.
• Biseparable: at least one bipartition is unentangled, but not all of them. Some genuine entanglement exists, but it doesn't involve every qubit at once.
• Genuinely multipartite entangled: every possible bipartition shows entanglement. No qubit can be peeled off as an independent, unentangled piece — the entanglement is fundamentally a property of the whole group.

"Genuinely entangled" doesn't mean "the same kind of entangled"

This is the chapter's most important warning, and it's exactly why Ch. 17 needs to exist at all: passing the "genuinely multipartite entangled" test in Fig. 16.1 is necessary but nowhere near sufficient to characterize a state's entanglement. GHZ and W both pass — every bipartition entangled for both — and yet Ch. 15 already told you they're LOCC-inequivalent: no sequence of local operations and classical communication can convert one into the other. The bipartition test in this chapter is a coarse classification (3 categories); a complete description of multipartite entanglement structure requires much finer tools, several of which (entanglement entropy, the various tangle measures) are active areas of ongoing research, not settled textbook material.