| GHZ state | |
|---|---|
| Name | GHZ state |
| Type | Multipartite entangled state |
| Introduced | 1989 |
| Developers | Daniel M. Greenberger, Michael A. Horne, Anton Zeilinger |
| Applications | Quantum information, Quantum computing, Quantum cryptography |
GHZ state
The GHZ state is a class of quantum states representing maximally entangled multipartite systems, most famously for three qubits. First introduced by Daniel M. Greenberger, Michael A. Horne and Anton Zeilinger in 1989, GHZ states provide sharp demonstrations of quantum nonlocality and are central to protocols in quantum information and quantum computing.
The prototypical GHZ state for three two-level systems (qubits) is the pure state |GHZ_3⟩ = (|000⟩ + |111⟩)/√2, expressed in the computational basis of qubits. More generally, an n-qubit GHZ state is |GHZ_n⟩ = (|0...0⟩ + |1...1⟩)/√2, which is an example of a stabilizer state in the stabilizer formalism. The GHZ state contrasts with the W state by its entanglement structure: GHZ entanglement is genuinely multipartite and can be destroyed by tracing out a single subsystem, whereas W-class states retain bipartite entanglement after loss of one qubit. The GHZ state is also a resource state for GHZ-type proofs of nonlocality and for certain quantum error correction codes such as Shor code building blocks.
Preparation methods depend on the physical platform. In trapped ion systems, GHZ states have been generated via entangling gates like the Mølmer–Sørensen gate using ions in electromagnetic traps at NIST and university laboratories. In superconducting qubit circuits, tunable couplers and CNOT or CZ gate sequences create GHZ states on processors by companies and institutions such as Google (company), IBM, and academic groups. Photonic implementations use spontaneous parametric down-conversion and linear optics elements with interferometers demonstrated by groups in University of Innsbruck and by Anton Zeilinger's group; these realizations exploit polarization or path encoding. Other implementations include neutral atom arrays with Rydberg interactions, nitrogen-vacancy center ensembles in diamond, and cavity quantum electrodynamics setups at institutions such as Max Planck Institute for Quantum Optics.
GHZ states exemplify genuine multipartite entanglement: the state is not separable with respect to any bipartition. They are eigenstates of collective operators in the Pauli matrices algebra (e.g., X⊗X⊗X and Z⊗Z⊗I), forming a simple stabilizer group. Entanglement measures applied to GHZ states include three-tangle and multipartite concurrence; the three-tangle for GHZ_3 is nonzero, distinguishing it from the W state class. GHZ correlations violate local realism maximally, producing deterministic contradictions with local hidden variable models rather than statistical inequalities. In resource theories, GHZ states are resources for tasks requiring global phase correlations and can be converted under LOCC only into limited forms of entanglement.
The GHZ paradox provides an all-or-nothing conflict between the predictions of quantum mechanics and those of local hidden variable theories without relying on inequalities like Bell's theorem. The original GHZ argument uses perfect correlations of three qubits to derive contradictory deterministic assignments under locality assumptions. Experimental demonstrations of GHZ nonlocality have been reported in photonic experiments (e.g., using high-fidelity state tomography), trapped ions, and superconducting circuits; such tests are complementary to Bell test experiments and have played roles in foundational studies by researchers such as John S. Bell and Zeilinger. Variations include tests of contextuality related to the Kochen–Specker theorem and multipartite loophole-free nonlocality tests.
GHZ states are used in quantum communication protocols such as quantum secret sharing, quantum teleportation networks, and entanglement swapping. They serve as resources for measurement-based quantum computation when embedded in graph states or cluster-state architectures. In quantum metrology, GHZ states enable phase-estimation protocols that can reach the Heisenberg limit under ideal conditions, improving sensitivity over separable states. They also play roles in multipartite quantum key distribution schemes and are ingredients in error-detection primitives for fault-tolerant quantum computing and quantum error correction codes developed at institutions like Harvard University and MIT.
Generalizations include GHZ states for higher-dimensional systems (qudits), asymmetric GHZ-like states with unequal amplitudes, and mixed-state analogues such as Werner–GHZ mixtures. Theoretical frameworks connect GHZ states to graph states and stabilizer codes; for example, an n-qubit GHZ state is LU-equivalent to a star graph state. Variants include cluster-GHZ hybrids, decoherence-free-subspace encodings, and continuous-variable GHZ analogues in quantum optics using squeezed states. Studies in entanglement classification characterize GHZ-class states under stochastic LOCC (SLOCC) equivalence and relate them to invariants like hyperdeterminants.
GHZ states are highly sensitive to noise and loss: decoherence mechanisms such as amplitude damping, phase damping, and particle loss rapidly degrade multipartite coherence and entanglement. Scaling GHZ preparation to large n faces exponential fragility and requires high-fidelity gates, error mitigation, and entanglement purification protocols. Practical experiments contend with detector inefficiencies, timing jitter in photonic setups, crosstalk in superconducting qubits, and motional heating in trapped ions. Overcoming these limitations involves advances in quantum control, cryogenics, high-efficiency single-photon detectors (e.g., superconducting nanowire detectors developed by groups at NIST and industry partners), and error-correcting architectures proposed by researchers such as Peter Shor and Alexei Kitaev.
Category:Quantum states Category:Quantum information theory Category:Entanglement