GHZ states and correlations
A GHZ state is a group of qubits in a superposition of "all zeros" and "all ones." Every measurement gives a string where all the bits agree, like 000 or 111, even though each qubit alone acts like a fair coin. You build it with one H gate and a chain of CX gates. Because it breaks so easily, it is a standard stress test for real quantum hardware.
How do you build a GHZ state?
GHZ is named after three physicists: Greenberger, Horne and Zeilinger. The recipe grows from the Bell pair, one CX gate at a time. (A CX flips its target qubit when its control qubit is 1.)
- H on q0 opens a superposition.
- A CX from q0 to q1 brings the second qubit into it.
- A CX from q1 to q2 brings in the third.
With three qubits, the state is (|000⟩ + |111⟩)/√2. That is a GHZ state. It has one part where every qubit reads 0 and one where every qubit reads 1. There is nothing else.
Like the Bell pair, every mixed answer has an amplitude of exactly zero. (An amplitude is the number that sets how likely an answer is. Square it to get the chance.) So each shot should read 000 or 111. All three bits agree, every time. Yet each qubit on its own acts like a fair coin.
Let's check the odds. Each of the two parts has amplitude 1/√2. Square it: (1/√2)² = ½. So about 500 of 1,000 shots should read 000, and about 500 should read 111.
Does the pattern keep working with more qubits?
Nothing in the recipe stops at three. Add two more CX gates and five qubits share one superposition: (|00000⟩ + |11111⟩)/√2.
Let's count them. Five bits can make 2 × 2 × 2 × 2 × 2 = 32 different strings. The ideal state uses exactly two of those 32.
Stop and notice how odd that is. Measure any one qubit and you get a fair coin. Measure all five and the coins have always agreed, shot after shot. Picture five friends in five cities each flipping a coin, and all five always landing the same. Unlike magic coins, though, you can't use this to send a message. No normal setup can copy this match along more than one measurement axis. It is the same argument as for the Bell pair, just bigger.
Why are GHZ states fragile?
A GHZ state is a superposition of two parts that are as different as they can be. That makes it very easy to upset. A phase error is a small nudge to the sign or angle of an amplitude. A phase error on any one of the n qubits shifts the phase of the whole superposition. So the link between the two parts fades about n times faster than it would for one qubit. Bigger GHZ states fall apart faster, by design. Think of a long row of dominoes. The more dominoes you add, the more places one can be knocked over by accident.
Be careful about what the chart shows, though. When you measure 0 versus 1, you only see bit-flip errors and readout errors. They show up as near-miss strings like 00100 or 11011. Phase damage doesn't show up here at all. A GHZ state that has lost its link completely still shows the same two tall bars. To prove the superposition is really alive, you have to measure along other axes too. These tests are called parity-oscillation experiments. They are exactly how hardware teams check large GHZ states.
How big can GHZ states get on real hardware?
GHZ circuits are a standard stress test because they break so easily. Each extra qubit adds another two-qubit gate, more depth (more steps in a row), and one more qubit that can lose its state. On a noisy device, the two tall bars stay clear for small GHZ states, with near-miss strings growing around them. Make it bigger and the pattern washes out.
Truly checked GHZ entanglement has been shown at the scale of a few dozen qubits on trapped-ion and superconducting machines. "Truly checked" means tested along more than one axis, not just read off the chart. That shows how good a device is. It is not a practical computing ability. Keeping those two apart is what the hardware comparison is for. Build the three- and five-qubit versions yourself in the lab.