Quantum Cookbook

Part III · Chapter 17

GHZ and W States

Read ~20 min Prerequisites Ch. 16 Tools qubit-loss simulator, code cell Live
Basic

Ch. 16 left a cliffhanger: two states can both be "genuinely multipartite entangled" and still be entangled in completely different ways. GHZ and W states are the two canonical examples — same pass/fail result on Ch. 16's test, opposite behavior on the single most practically important question you can ask about an entangled state: what happens if you lose one of the particles?

The two states

|GHZ_n⟩ = (|00...0⟩ + |11...1⟩)/√2

|W_n⟩ = (1/√n)(|10...0⟩ + |01...0⟩ + ... + |00...1⟩)

GHZ is "all qubits agree" — every term has every qubit the same. W is "exactly one qubit is excited, we just don't know which" — an equal superposition over every possible location of a single 1.

Intermediate
Fig. 17.1 — pick a state, lose a qubit, inspect what's left3-qubit case
—
Lose any qubit from GHZ₃: the remaining two collapse to a classical 50/50 mixture of |00⟩ and |11⟩ — zero quantum coherence left, fully separable. Lose any qubit from W₃: the remaining two retain genuine entanglement (the off-diagonal terms in the matrix above don't vanish) — W survives where GHZ collapses completely.

Why this happens

GHZ's "all-or-nothing" correlation is exactly its weakness: knowing any one qubit's value tells you everything about the rest, which also means losing access to any one qubit destroys everything you could have known. W's correlation is spread out — "exactly one qubit is excited" remains a meaningful, partially-known fact even if you can't see one of the qubits, because the excitation might have been on the qubit you lost, or might not have been.

Advanced

This is provably not just a different formula — it's a different resource

Ch. 15 mentioned that GHZ and W states are LOCC-inequivalent. The qubit-loss behavior in Fig. 17.1 is the most direct intuition for why: any LOCC protocol converting GHZ into W (or vice versa) would have to preserve robustness-to-loss in some convertible way, but GHZ's entanglement vanishes completely on qubit loss while W's only degrades — a structural difference no amount of local processing and classical chat can paper over. This isn't just a textbook curiosity: GHZ states are the right choice when you need maximal correlation and can guarantee no losses (e.g. some quantum cryptography protocols), while W states are preferred in noisy or lossy settings — like photonic systems, where a photon failing to arrive is a routine occurrence, not a rare failure.

Fig. 17.2 — code, runs in your browser
Compare the off-diagonal entries between the two printed matrices directly — that comparison is the entire content of this chapter, made numerically explicit.
Check your understanding

Write down the 3-qubit GHZ and W states.

GHZ₃ = (|000⟩+|111⟩)/√2. W₃ = (|100⟩+|010⟩+|001⟩)/√3.

Check your understanding

What happens to the entanglement of the remaining qubits if you lose one qubit from a GHZ state?

It vanishes completely — the remaining qubits become a classical, separable 50/50 mixture with no quantum coherence left.

Check your understanding

Why does W survive qubit loss when GHZ doesn't?

GHZ's correlation is all-or-nothing — every qubit's value reveals all the others, so losing one destroys the whole structure. W's "exactly one qubit excited" fact stays meaningful even without seeing every qubit, since the excitation might or might not have been on the lost one.

Check your understanding

What does "LOCC-inequivalent" mean for GHZ and W, and what evidence from this chapter supports it?

It means no LOCC protocol can convert one into the other. Their opposite behavior under qubit loss (GHZ collapses entirely, W only degrades) is a structural difference that any conversion would have to somehow preserve and can't — strong intuition for why they're inequivalent.

Check your understanding

Why might a real photonic experiment prefer W states over GHZ states?

Photon loss is routine in real hardware, not rare. W states retain useful entanglement after a loss event; GHZ states lose everything, making W the more practical choice whenever loss is a realistic concern.

Review queue

Scroll back to a chapter's quiz cards to review them once they're due — a full cross-chapter queue view is a planned upgrade.