One last surprise to close out Part III: the W state you spent all of Ch. 17 studying wasn't actually its own separate thing. It's the simplest member of a whole family — Dicke states — and Part III's two headline examples, GHZ and W, turn out to sit in very different relationships to that family.
The definition
The Dicke state |D_n^k⟩ is the equal superposition of every n-qubit bitstring with exactly k ones (Hamming weight k):
|D_n^k⟩ = C(n,k)^(−1/2) Σ_{|x|=k} |x⟩
where C(n,k) is "n choose k," the number of distinct ways to place k ones among n positions, and the sum runs over every one of those bitstrings.
IntermediateWhere GHZ fits — and doesn't
W = D_n^1 is a Dicke state by definition. GHZ is not a Dicke state at all — it's a superposition of two different Dicke states, the trivial k=0 and k=n cases:
|GHZ_n⟩ = (|D_n^0⟩ + |D_n^n⟩)/√2 = (|00...0⟩ + |11...1⟩)/√2
This is a clean, formal way to state something Ch. 17 only showed you experimentally: W sits inside the well-behaved, permutation-symmetric Dicke family in the most direct way possible, while GHZ straddles two trivial corners of that same family.
AdvancedWhy "permutation symmetric" is the deeper property
Every Dicke state is invariant under relabeling which qubit is which — swap any two qubits' positions and the state is completely unchanged, since the definition only ever cares about the count of ones, never their positions. This permutation symmetry isn't a curiosity: Dicke states are exactly the eigenstates of total collective angular momentum that show up throughout atomic and condensed-matter physics (the name comes from R.H. Dicke's 1954 work on collective spontaneous emission), and they're an active research tool in quantum metrology today, since a Dicke state's sensitivity to a parameter you're trying to measure can, in the right setup, beat what unentangled particles could ever achieve.
- ArticleDel Toro Barba — Dicke States in Quantum ComputingAccessible introduction confirming the definition and W-state special case used in this chapter.
- PaperBärtschi & Eidenbenz, "Deterministic Preparation of Dicke States" (arXiv)How to actually build a Dicke state with a real quantum circuit, beyond this chapter's definition.
- Paper"Sheaf-Theoretic Methods in Quantum Mechanics" (arXiv)Confirms the formal Dicke state definition and notes W as the n,1 special case explicitly.
End of Part III
Teleportation, LOCC, the separability hierarchy, GHZ/W, graph states, and Dicke states form a complete map of what "entangled" can mean once you're past two qubits. Part IV picks up the photonics thread Ch. 18 only gestured at — polarization beam splitters and the fusion gates that actually build graph states like the one in Fig. 18.1 out of real photons.