GHZ states and correlations

The Bell-pair recipe extends one CX at a time. H on q0 opens the superposition; a CX from q0 to q1 brings the second qubit into it; a CX from q1 to q2 brings in the third. After three qubits the state is (|000⟩ + |111⟩)/√2 — a GHZ state: one branch where every qubit reads 0, one where every qubit reads 1, and nothing else.

As with the Bell pair, every mixed outcome has amplitude exactly zero. Each shot should read 000 or 111 — all three bits agreeing, every time — even though each individual qubit, viewed alone, is a fair coin.

Nothing about the recipe stops at three. Chain another two CXs and five qubits share one superposition: (|00000⟩ + |11111⟩)/√2. There are thirty-two possible outcome strings, and the ideal state uses exactly two of them.

It's worth pausing on how strange that is: measure any one qubit and you get a fair coin; measure all five and the coins have always agreed, shot after shot. No classical mechanism reproduces that across more than one measurement basis — the same argument as for the Bell pair, scaled up.

A GHZ state is a superposition of two maximally different branches, and that makes it exceptionally sensitive. A phase error on any one of the n qubits shifts the relative phase of the whole superposition, so the coherence between the branches decays roughly n times faster than a single qubit's. Bigger GHZ states die faster, by construction.

Be precise about what the histogram shows, though. In the computational basis you only see bit-flip and readout errors — they appear as near-miss strings like 00100 or 11011. Phase damage is invisible here: a GHZ state that has completely lost its coherence still shows the same two tall bars. Certifying that the superposition is genuinely alive requires measurements in other bases (parity-oscillation experiments), which is exactly how hardware teams verify large GHZ states.

GHZ circuits are a standard stress test precisely because they are fragile: each extra qubit adds another two-qubit gate, more depth, and another qubit's worth of decoherence exposure. On a noisy device the two tall bars stay visible for small GHZ states, with near-miss strings growing around them; push the size up and the structure washes out.

Genuinely verified GHZ entanglement — checked in multiple bases, not just read off the histogram — has been demonstrated at the scale of a few dozen qubits on trapped-ion and superconducting machines. That is a demonstration of device quality, not a practical computing capability, and keeping that distinction straight is what the hardware comparison is for. Build the 3- and 5-qubit versions yourself in the lab.