History of
The Gerlach-Steim State
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+---
+title: The Gerlach-Steim State
+updated: 2026-09-05
+updated_at: 2026-09-05T14:23:44.362Z
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+updated_ip: visitor-99c4
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+---
+# 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:
+
+1. **The GHZ class** — maximally entangled, but fragile. Measure one qubit in the computational basis and the entanglement is destroyed.
+2. **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.
+
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