GHZ State
What You'll Learn: How to extend quantum entanglement beyond two qubits, creating an "all-or-nothing" correlation that proves quantum mechanics cannot be explained by any local hidden variable theory — without needing statistics.
Level: Beginner | Time: 15 minutes | Qubits: 3 | Framework: Qiskit
Prerequisites: Bell State
The Idea
In a Bell state, two qubits always agree. A GHZ state takes this further: three qubits always agree — they're either all 0 or all 1, with nothing in between.
Why does this matter? In 1989, Greenberger, Horne, and Zeilinger showed that a single measurement on a GHZ state can prove quantum mechanics is fundamentally non-classical. While Bell's theorem needs many measurements and statistical analysis, the GHZ argument works with a single shot — a "Bell's theorem without inequalities."
But there's a catch: GHZ entanglement is fragile. If you lose even one of the three qubits (it drifts away or decoheres), the remaining two are left in a completely unentangled state. This makes GHZ states both powerful for tests of quantum mechanics and challenging for real-world applications.
How It Works
The Circuit
CODE┌───┐ q_0: ┤ H ├──■────■── └───┘┌─┴─┐ │ q_1: ─────┤ X ├──┼── └───┘┌─┴─┐ q_2: ──────────┤ X ├ └───┘
Step 1: Superposition
PYTHONqc.h(0) # Put qubit 0 into superposition: (|0⟩ + |1⟩) / √2
Just like the Bell state, we start by putting qubit 0 into an equal superposition. The other two qubits remain in |0⟩.
State after Step 1: (|0⟩ + |1⟩)/√2 ⊗ |0⟩ ⊗ |0⟩ = (|000⟩ + |100⟩)/√2
Step 2: First Entanglement
PYTHONqc.cx(0, 1) # If qubit 0 is |1⟩, flip qubit 1
The first CNOT correlates qubit 1 with qubit 0 — identical to the Bell state step.
State after Step 2: (|000⟩ + |110⟩)/√2
Step 3: Second Entanglement
PYTHONqc.cx(0, 2) # If qubit 0 is |1⟩, flip qubit 2
The second CNOT brings qubit 2 into the entangled group. Now all three qubits are correlated.
State after Step 3: (|000⟩ + |111⟩)/√2 = |GHZ⟩
The Pattern
To create an N-qubit GHZ state, you always use the same recipe:
- Hadamard on qubit 0
- CNOT from qubit 0 to each of qubits 1, 2, ..., N-1
This means the circuit depth grows linearly with N. On hardware with only nearest-neighbor connectivity, you'd need swap gates — but on simulators, any qubit can control any other.
The Math
State Evolution
Starting from |000⟩:
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Initial: |ψ₀⟩ = |000⟩
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After H: |ψ₁⟩ = (H ⊗ I ⊗ I)|000⟩ = (|000⟩ + |100⟩)/√2
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After CX(0,1): |ψ₂⟩ = (|000⟩ + |110⟩)/√2
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After CX(0,2): |ψ₃⟩ = (|000⟩ + |111⟩)/√2 = |GHZ₃⟩
General N-Qubit Form
|GHZ_N⟩ = (|0⟩^⊗N + |1⟩^⊗N) / √2
For N = 2 this is exactly the Bell state |Φ+⟩.
Key Property: Monogamy of Entanglement
The GHZ state has maximum multipartite entanglement but zero bipartite entanglement. If you trace out any one qubit, the remaining qubits are in a mixed state with no entanglement:
ρ₁₂ = Tr₃(|GHZ⟩⟨GHZ|) = (|00⟩⟨00| + |11⟩⟨11|) / 2
This is a classical mixture — not a quantum superposition. The entanglement truly lives in the three-body correlation, not in any pair.
GHZ vs. W State
There are different "kinds" of three-qubit entanglement:
| Property | GHZ: (|000⟩+|111⟩)/√2 | W: (|001⟩+|010⟩+|100⟩)/√3 | |----------|------------------------|----------------------------| | Lose 1 qubit | All entanglement lost | Remaining 2 still entangled | | Measurement correlations | All-or-nothing | Exactly one qubit is |1⟩ | | Bell violation | Maximum (deterministic proof) | Partial | | Use case | Quantum secret sharing, metrology | Robust quantum networks |
These represent fundamentally different entanglement classes — you cannot convert a GHZ state into a W state using only local operations.
Expected Output
When you measure all 3 qubits 1024 times:
| Outcome | Expected Count | Probability |
|---|---|---|
| |000⟩ | ~512 | ~50% |
| |111⟩ | ~512 | ~50% |
| All others | 0 | 0% |
All three qubits always agree — that's the signature of GHZ entanglement.
On real hardware: Decoherence breaks GHZ correlations faster than Bell state correlations. You'll see "leakage" into states like |001⟩, |010⟩, |100⟩, etc. The fraction of correct outcomes (|000⟩ + |111⟩) measures the hardware's multi-qubit fidelity.
Running the Circuit
PYTHONfrom circuit import run_circuit, verify_ghz_state # Basic execution — 3-qubit GHZ counts = run_circuit(shots=1024) print(counts) # {'000': ~512, '111': ~512} # Scale to 5 qubits counts = run_circuit(shots=1024, n_qubits=5) print(counts) # {'00000': ~512, '11111': ~512} # Verification — checks all-or-nothing correlations result = verify_ghz_state(n_qubits=3, shots=4096, tolerance=0.05) print(result["passed"]) # True for check in result["checks"]: print(f" {check['name']}: {check['detail']}")
Try It Yourself
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Scale up: Create a 5-qubit, then 10-qubit GHZ state. Are the probabilities still 50/50? (On a simulator, yes — on real hardware, fidelity drops with N.)
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Measure in X basis: Add
qc.h(i)before measurement on all qubits. What outcomes do you see? (Hint: it's related to GHZ's parity properties.) -
Simulate decoherence: Use Qiskit's noise model to add single-qubit depolarizing noise. How quickly does the |000⟩+|111⟩ fraction drop?
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Build a W state: Create (|001⟩ + |010⟩ + |100⟩)/√3 instead. Compare what happens when you trace out one qubit.
What's Next
- Quantum Teleportation — Use entanglement to transfer a quantum state
- Superdense Coding — Send 2 classical bits using 1 entangled qubit
- Bell State — Review the 2-qubit foundation
Applications
| Application | How GHZ States Are Used |
|---|---|
| Quantum Secret Sharing | Split a secret among N parties such that all must cooperate to recover it |
| Quantum Metrology | Heisenberg-limited phase estimation (√N improvement over classical) |
| Quantum Error Correction | Repetition codes encode a logical qubit in GHZ-like states |
| Bell Tests | Mermin's inequality gives exponentially increasing violation with N |
| Quantum Networks | Multipartite entanglement distribution for quantum internet protocols |
References
- Greenberger, D.M., Horne, M.A., Zeilinger, A. (1989). "Going Beyond Bell's Theorem." In: Bell's Theorem, Quantum Theory and Conceptions of the Universe, pp. 69-72. arXiv: 0712.0921
- Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A. (1990). "Bell's theorem without inequalities." American Journal of Physics 58(12), 1131-1143.
- Mermin, N.D. (1990). "Extreme quantum entanglement in a superposition of macroscopically distinct states." Physical Review Letters 65(15), 1838.
- Nielsen, M.A. & Chuang, I.L. Quantum Computation and Quantum Information, Section 1.3.6.