Multipartite Entanglement

With three or more parties, entanglement stops being a single axis. There is no Schmidt decomposition for three subsystems, no unique "amount," and — most strikingly — inequivalent kinds of entanglement that cannot be converted into one another even by stochastic local operations. This lesson maps the landscape.

Separability has many layers

For $N$ parties, a pure state can be entangled across some cuts and not others. The classification starts with partitions:

Fully product: $|\psi\rangle = |\phi_1\rangle\otimes\cdots\otimes|\phi_N\rangle$ — no entanglement at all.
Biseparable: the state factorizes across some bipartition into two groups, e.g. $|\psi\rangle_{ABC} = |\phi\rangle_A \otimes |\chi\rangle_{BC}$ , even if it is entangled within a group. A mixed state is biseparable if it is a convex mixture of states each separable across (possibly different) bipartitions.
Genuinely multipartite entangled (GME): entangled across every bipartition — it cannot be written as a mixture of states that each factor across some cut. This is the strongest, "all parties truly share" form.

Detecting GME is harder than bipartite entanglement: a state can be entangled across each individual cut yet still be biseparable as a mixture, so checking cuts one at a time is not enough.

SLOCC classes: the three-qubit dichotomy

To classify kinds of entanglement we relax LOCC to SLOCC — stochastic LOCC, where we only require the transformation to succeed with nonzero probability. Two states are SLOCC-equivalent when each can be turned into the other by invertible local operators,

|\psi\rangle = (A_1 \otimes A_2 \otimes \cdots \otimes A_N)\,|\phi\rangle, \qquad A_k \in \mathrm{GL}(d,\mathbb{C}),

up to normalization. SLOCC-equivalent states can perform the same tasks (with some success probability), so SLOCC classes are the natural "types" of entanglement.

Dür, Vidal, and Cirac proved the headline result: a genuinely-entangled three-qubit pure state belongs to exactly one of two SLOCC classes, with no invertible local map between them:

|\mathrm{GHZ}\rangle = \tfrac{1}{\sqrt2}\big(|000\rangle + |111\rangle\big), \qquad |\mathrm{W}\rangle = \tfrac{1}{\sqrt3}\big(|001\rangle + |010\rangle + |100\rangle\big).

Together with the fully-product and the three biseparable families, three qubits fall into six SLOCC classes in total — but only two of them are genuinely tripartite. The next lesson contrasts GHZ and W in detail.

More parties, more classes — and a flood

The clean two-class picture is special to three qubits. As soon as you go to four qubits, SLOCC gives infinitely many inequivalent classes (organized into nine families by Verstraete et al.). The reason is parameter counting: the number of SLOCC invariants grows faster than the number of local degrees of freedom you can use to remove them, so continuous families of inequivalent states appear. Multipartite entanglement is genuinely richer than bipartite, not merely "more of the same."

How we measure it anyway

Because no single number works, several complementary tools are used:

Bipartite-cut entropies. Compute $S(\rho_A)$ across each bipartition; this profiles the entanglement structure but cannot certify GME by itself.
Tangle / three-tangle. For three qubits the three-tangle $\tau_{ABC}$ is a genuine tripartite invariant: it is nonzero for GHZ and exactly zero for W, quantitatively separating the two classes (and central to the monogamy relation of the next-but-one lesson).
GME measures and witnesses. Genuine-multipartite-entanglement measures (e.g. the GME concurrence) and entanglement witnesses detect and bound entanglement across all cuts at once.