Multiparty Entanglement

Bell pairs entangle two parties, but many network applications — conference key agreement, distributed sensing, anonymous broadcasting, leader election — need three or more nodes sharing correlations at once. The workhorse multiparty resource is the GHZ state, named after Greenberger, Horne and Zeilinger.

The GHZ state

For $n$ parties, each holding one qubit, the GHZ state is

|\mathrm{GHZ}_n\rangle = \frac{|0\rangle^{\otimes n} + |1\rangle^{\otimes n}}{\sqrt{2}}.

For $n = 3$ that is $\tfrac{1}{\sqrt 2}\big(|000\rangle + |111\rangle\big)$ . It is the natural generalisation of the Bell pair $\tfrac{1}{\sqrt2}(|00\rangle + |11\rangle) = |\mathrm{GHZ}_2\rangle$ : all parties are either $0$ or all are $1$ , with nothing in between.

Building it on a chain

A GHZ state is prepared with one Hadamard followed by a cascade of CNOTs:

|\mathrm{GHZ}_n\rangle = \mathrm{CNOT}_{n-2\to n-1}\cdots \mathrm{CNOT}_{0\to 1}\,(H\otimes I^{\otimes n-1})\,|0\rangle^{\otimes n}.

The Hadamard seeds the superposition $\tfrac{1}{\sqrt2}(|0\rangle + |1\rangle)$ on the first qubit; each CNOT then forces the next qubit to agree with the running value, propagating the all- $0$ /all- $1$ branch structure across the whole register. Across a network, this can be done by a source that fans out CNOTs, or assembled from Bell pairs and fusion operations at a central node and then distributed.

A telling fragility

GHZ entanglement is powerful but brittle. Measure (or lose) a single qubit of $|\mathrm{GHZ}_n\rangle$ in the computational basis and the entanglement of the remaining $n-1$ qubits collapses entirely — they are left in a plain product state. Contrast this with a different multiparty class, the W state $\tfrac{1}{\sqrt n}(|10\cdots0\rangle + \cdots + |0\cdots01\rangle)$ , whose pairwise entanglement survives the loss of a party. GHZ and W are inequivalent: no sequence of local operations and classical communication can turn one into the other. The right multiparty resource depends on whether your application needs all-or-nothing correlation (GHZ) or robustness to dropout (W).

Correlations across bases

Like the Bell pair, the GHZ state hides its entanglement from any single party: each qubit on its own is maximally mixed. In the $Z$ basis the parties see perfect agreement ( $000$ or $111$ , each with probability $\tfrac12$ ). In the $X$ basis the parity of the outcomes is fixed — the source of the celebrated GHZ contradiction with local hidden variables, which needs no inequalities and no statistics: a single run can refute local realism. That same parity structure is what multiparty protocols measure to detect eavesdroppers or to extract shared randomness.

Try it

Build a 3-party GHZ state across Alice, Bob and Charlie, then measure. The grader checks the distribution: only 000 and 111 should appear, each with probability $\tfrac12$ .

Run your code to see the quantum state.

In the Probabilities tab you should see exactly two bars of height $0.5$ — the multiparty all-or-nothing signature.