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GHZ State

The GHZ (Greenberger-Horne-Zeilinger) state is a maximally entangled multi-qubit state of the form (|000...0⟩+|111...1⟩)/sqrt(2) that demonstrates non-classical correlations stronger than Bell states.

What it means

The GHZ state generalizes Bell state entanglement to three or more qubits.For three qubits: |GHZ⟩ = (|000⟩+|111⟩)/sqrt(2).It is created by applying H to the first qubit followed by CNOT gates cascading to each subsequent qubit.GHZ states exhibit 'all-or-nothing' correlations: measuring any single qubit collapses all others.GHZ states are more fragile than Bell states -- losing a single qubit destroys all entanglement.They are central to quantum secret sharing, quantum error correction, and tests of quantum non-locality without inequalities.

Everyday analogy

A GHZ state is like a group of synchronized swimmers who all flip simultaneously -- they are perfectly coordinated, but if one swimmer is removed, the coordination of the entire group is broken.
Think of a house of cards -- all cards depend on each other, and removing one collapses the entire structure.
The GHZ circuit (one H plus a chain of CNOTs) is the quantum computer's fitness test: ideally only 000... or 111... should ever appear. How many qubits a machine can entangle at once — and how cleanly — is a whole-device benchmark, and any asymmetry in the measured histogram directly exposes calibration quality.

Common misconceptions

  • GHZ entanglement is NOT just pairwise -- it is genuinely multipartite; no pair of qubits is entangled if the third is traced out.
  • GHZ states are NOT more robust than Bell states -- they are actually more fragile, as losing one qubit destroys all entanglement.
  • Building a large GHZ state sequentially makes CNOT depth grow linearly — and since every CNOT takes time while decoherence keeps accumulating, naive construction fails; real benchmarks parallelize CNOTs to reduce circuit depth.

Key takeaways

  • |GHZ⟩ = (|000...0⟩+|111...1⟩)/sqrt(2) is maximally entangled across all qubits.
  • Created by H + cascading CNOTs.
  • Demonstrates genuinely multipartite entanglement, distinct from pairwise entanglement.
  • GHZ preparation is a standard whole-device entanglement benchmark; IBM reported 120-qubit GHZ preparation in 2025 using optimized CNOT placement (as of 2026).

Check your understanding

What happens to a 3-qubit GHZ state if one qubit is lost (traced out)?

  1. A.A Bell state remains
  2. B.A product state remains
  3. C.A maximally mixed state of the remaining 2 qubits
  4. D.Nothing changes
Show the answer

Answer: C. A maximally mixed state of the remaining 2 qubits

Why: Tracing out one qubit of a GHZ state yields a maximally mixed state (no entanglement) for the remaining two qubits -- GHZ entanglement is genuinely tripartite.

Builds on

QubitQuantum EntanglementBell StatesHadamard GateCNOT Gate

Primary source: Greenberger, Horne, Shimony & Zeilinger, Am. J. Phys. 58, 1131 (1990), doi:10.1119/1.16243

IBM 120-qubit GHZ benchmark sentence in keyTakeaways is explicitly as-of-2026 qualified.

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