The Gerlach-Steim State
Field Notes on Tripartite Entanglement
Beyond Two
Two-particle entanglement is a parlor trick compared to what happens when you bring in a third.
The GHZ state — named after Greenberger, Horne, and Zeilinger, 1989 — is the simplest multipartite entangled state:
|GHZ⟩ = (|000⟩ + |111⟩) / √2
Three qubits. Either all zero or all one. A perfect superposition of both. The state is irreducible — it cannot be factored into a product of independent single-particle states. Measure any one qubit, and the other two instantly collapse. Not just to correlated values. To identical values.
The GHZ Paradox
Here is where two particles lose their parity.
For the singlet state (two particles), Bell's inequality shows the violation through statistics. You need many runs. Many measurements. A statistical average.
For the GHZ state: one measurement suffices.
Prepare three qubits in the GHZ state. Alice measures along X. Bob measures along Y. Charlie measures along Y.
Quantum mechanics predicts the result with certainty — not probability, not correlation. Certainty. The outcome is always −1.
A local hidden variable model predicts +1.
Contradiction in a single shot. No statistics. No error bars. Just one measurement and the theory is dead.
This is not a subtle violation. This is a knife.
Experimental Realization
1999 — Bouwmeester et al.: First GHZ state created experimentally using spontaneous parametric down-conversion. Three photons, entangled in polarization. The violation matched quantum prediction.
2004 — Pan et al.: Three-particle GHZ entanglement over increasing distances. Proof that multipartite entanglement is not confined to the lab bench.
2010s — Ion traps and superconducting qubits: GHZ states with many more particles. The scaling is difficult — each additional qubit multiplies the coherence requirements. But it has been done. Sixteen qubits in an ion trap. The entanglement survives long enough for computation.
The Structure
Multipartite entanglement has structure that two-particle entanglement lacks. Some states are more entangled than others. The entanglement can be distributed differently — one qubit maximally entangled with the rest, or distributed more evenly. This leads to the classification of entanglement under local unitary transformations.
Two inequivalent classes of three-qubit entanglement:
- The GHZ class — maximally entangled, but fragile. Measure one qubit in the computational basis and the entanglement is destroyed.
- The W class — more robust. Removing one qubit leaves the remaining two entangled.
|W⟩ = (|001⟩ + |010⟩ + |100⟩) / √3
Three qubits. Exactly one is 1. You don't know which one until you measure. But even if one qubit is lost, the remaining two retain entanglement. GHZ loses everything. W survives.
Nature offers options. Even in entanglement, there are strategies.
Practical Use
GHZ states are the workhorse of quantum error correction. The three-qubit bit-flip code uses a GHZ-like state to detect and correct errors without measuring (and thus destroying) the logical information. In quantum secret sharing, a GHZ state allows three parties to share a secret such that no two can recover it alone — all three must collaborate.
The universe is not content with pairwise entanglement. It wants to build structures. And GHZ is the smallest brick.