The Quantum Investor's Notebook logo The Quantum Investor's Notebook
← Back

Wednesday, September 16, 2026

The GHZ State: All Together, or Not at All

A GHZ stateA maximally entangled state of three or more qubits: an all-or-nothing superposition of all-zeros and all-ones. If one qubit is lost or ignored, the remaining GHZ entanglement is destroyed. puts three or more qubitsThe basic unit of a quantum computer. Like a 'bit' in a normal computer, but instead of being only 0 or 1 it can be 0, 1, or a blend of both at once. into one coherent all-or-nothing state. It is simple to write down, difficult to preserve, and useful because the entire register has to stay quantum together.

One state, not many matching bits

For N qubits, the Greenberger-Horne-Zeilinger (GHZ) state is1

|GHZN⟩  =  (|00…0⟩ + |11…1⟩) / √2

Measure every qubit as 0 or 1 and the result looks almost boring: either every qubit is 0 or every qubit is 1. That agreement is not what makes the state quantum, because a classical machine could flip one coin and copy its result into every bit. The difference is the definite relative phaseThe phase difference between the parts of a superposition. Invisible when you only ask 0 or 1, it changes the results when you measure another way. between the two branches, here the plus sign: the all-zero and all-one possibilities remain parts of one coherent quantum stateThe mathematical description used to predict the probabilities of different measurement outcomes for a quantum system..

Agreement is easy to fake. Coherence is not.

That phase belongs to the whole register. If one qubit is simply lost or ignored, the remaining qubits are left in a classical mixtureA system that is secretly in one definite state or another, picked at random. It can mimic some quantum statistics but carries no phase relationship. of all-zeros and all-ones rather than a smaller GHZ state. This is genuine multipartite entanglementEntanglement shared across every qubit in a group at once, not just among smaller subsets.: the quantum relationship cannot be reduced to an entangled pair plus spectators.2

How you build one

Start with four qubits in |0000⟩ and apply a Hadamard gateA one-qubit operation that turns a definite 0 or 1 into a balanced quantum superposition. The standard way to create one. to the first qubit, which creates two alternatives:

|0000⟩  →  (|0000⟩ + |1000⟩) / √2

Now use CNOT gatesA two-qubit gate that flips a target qubit only when a control qubit is 1. The standard two-qubit building block. to make the other qubits follow the first one without measuring which alternative is present. One CNOT at a time turns the second branch into |1100⟩, then |1110⟩, then |1111⟩:

(|0000⟩ + |1111⟩) / √2

A star of CNOTs from the first qubit is one convenient circuit, not a requirement of GHZ itself. On locally connected hardware the same state can be grown along a chain or tree; connectivityWhich qubits in a machine can directly interact with each other. More connectivity makes more algorithms possible. changes circuit depthThe number of sequential layers of gates in a quantum circuit. Greater depth generally means a longer computation and more exposure to noise. and routing cost, not the definition of the state.

Build it, then change the question. Left: a four-qubit circuit in which a Hadamard gate on q1 creates two alternatives and three CNOT gates from q1 to q2, q3 and q4 spread them coherently, giving (|0000⟩ + |1111⟩)/√2, one shared state. Right: on a fresh copy, choose one measurement. Measuring Z gives 0000 or 1111, which a classical mixture can fake. Measuring X gives outcomes such as ++++, ++−−, +−+− and −−−−, with an even number of minus signs only. Bottom: agreement is easy to fake; coherence is not.
Figure 1: Build it, then change the question. The Z test and the X test are alternatives, each run on a freshly prepared copy.

How do you know the coherence is there?

Seeing only 0000 and 1111 is not enough, because a classical mixtureA system that is secretly in one definite state or another, picked at random. It can mimic some quantum statistics but carries no phase relationship. can fake those statistics perfectly. So prepare fresh copies and change the measurement basisThe set of alternatives a quantum measurement is designed to distinguish.. For the + GHZ stateA maximally entangled state of three or more qubits: an all-or-nothing superposition of all-zeros and all-ones. If one qubit is lost or ignored, the remaining GHZ entanglement is destroyed., measuring every qubitThe basic unit of a quantum computer. Like a 'bit' in a normal computer, but instead of being only 0 or 1 it can be 0, 1, or a blend of both at once. in the X basis gives only outcomes whose signs multiply to +1, which with four qubits means an even number of minus signs. The classical mixture behaves differently: in the X basis it produces every sign pattern at random. The test is not merely whether the qubits agree, but whether the phase between the two branches survives when you ask the register a different question.

This same structure gave GHZ its historical importance. Carefully chosen measurementsA physical process that produces a classical outcome and updates the quantum state. In an ideal projective measurement, the state is left in an eigenstate associated with the observed outcome. produce an all-or-nothing contradiction between quantum mechanics and local predetermined answers, rather than only a statistical Bell inequalityA numerical limit on the correlations any 'local hidden variable' theory can produce. Quantum entanglement breaks it, which is how we know the link is real. violation,1 and the first three-photon GHZ state was observed in 1999.3

Why quantum computers care

GHZ states concentrate several hard hardware requirements into a simple circuit. As the state grows, every added entangling operation and qubit gives noise another opportunity to damage the global coherenceA phase relationship between quantum alternatives that can change later measurement outcomes., so large GHZ states probe multi-qubit control, coherence, entangling-gate errors, compilation and connectivityWhich qubits in a machine can directly interact with each other. More connectivity makes more algorithms possible..4

But GHZ is not a miniature universal quantum algorithm. The standard preparation uses only Hadamard and CNOT gatesA two-qubit gate that flips a target qubit only when a control qubit is 1. The standard two-qubit building block., so it is a stabilizer circuitA circuit made only of Clifford gates and standard measurements. The Gottesman-Knill theorem shows a classical computer can simulate such circuits efficiently. and can be simulated efficiently on a classical computer.5 Making an excellent GHZ state demonstrates high-quality multipartite entanglementA quantum link where two qubits' states become tied together, so acting on or measuring one affects the other. It is a key resource for quantum computing., not quantum computational advantage.

GHZ-like cat states nevertheless appear inside fault-tolerant quantum computing. They can serve as ancillasA helper qubit used during a computation, for example to check other qubits for errors without reading their data directly. for stabilizer measurementsA check that reads a joint property of several qubits, such as whether they agree, without revealing the information they store. It is the basic step of quantum error correction. during quantum error correctionTechniques that combine many shaky physical qubits into fewer reliable ones, so a long calculation stays correct., helping prevent one faulty ancilla interaction from spreading errors across many data qubitsThe basic unit of a quantum computer. Like a 'bit' in a normal computer, but instead of being only 0 or 1 it can be 0, 1, or a blend of both at once..6

A GHZ state is both a resource and a stress test.

GHZ statesA maximally entangled state of three or more qubits: an all-or-nothing superposition of all-zeros and all-ones. If one qubit is lost or ignored, the remaining GHZ entanglement is destroyed. also matter outside computation. In ideal noiseless quantum sensingUsing quantum effects to take ultra-precise measurements, such as atomic clocks or navigation and satellite sensors., their relative phaseThe phase difference between the parts of a superposition. Invisible when you only ask 0 or 1, it changes the results when you measure another way. can accumulate N times faster than that of one probe, enabling precision at the Heisenberg limitThe ultimate precision limit of measurement, where error shrinks as one over the number of particles N, beating the classical square-root scaling.. The catch is the same feature that made the state interesting: the information is global, so noise that damages the shared phase can erase the advantage quickly.7 A GHZ state is therefore both a resource and a stress test. Its lesson is not simply that many qubits can be entangled: sometimes the quantum information belongs to the whole register at once.

Sources

  1. Greenberger, D. M., Horne, M. A., Shimony, A. & Zeilinger, A. “Bell’s Theorem Without Inequalities.” American Journal of Physics 58, 1131–1143 (1990). DOI: 10.1119/1.16243. ↩
  2. Dür, W., Vidal, G. & Cirac, J. I. “Three Qubits Can Be Entangled in Two Inequivalent Ways.” Physical Review A 62, 062314 (2000). DOI: 10.1103/PhysRevA.62.062314. ↩
  3. Bouwmeester, D., Pan, J.-W., Daniell, M., Weinfurter, H. & Zeilinger, A. “Observation of Three-Photon Greenberger-Horne-Zeilinger Entanglement.” Physical Review Letters 82, 1345–1349 (1999). DOI: 10.1103/PhysRevLett.82.1345. ↩
  4. Bao, Z., Xu, S., Song, Z. et al. “Creating and Controlling Global Greenberger-Horne-Zeilinger Entanglement on Quantum Processors.” Nature Communications 15, 8823 (2024). DOI: 10.1038/s41467-024-53140-5. ↩
  5. Aaronson, S. & Gottesman, D. “Improved Simulation of Stabilizer Circuits.” Physical Review A 70, 052328 (2004). DOI: 10.1103/PhysRevA.70.052328. ↩
  6. Prabhu, P. & Reichardt, B. W. “Fault-Tolerant Syndrome Extraction and Cat State Preparation with Fewer Qubits.” Quantum 7, 1154 (2023). DOI: 10.22331/q-2023-10-24-1154. ↩
  7. Giovannetti, V., Lloyd, S. & Maccone, L. “Advances in Quantum Metrology.” Nature Photonics 5, 222–229 (2011). DOI: 10.1038/nphoton.2011.35. ↩