| Greenberger–Horne–Zeilinger state | |
|---|---|
| Name | Greenberger–Horne–Zeilinger state |
| Introduced | 1989 |
| Authors | Greenberger, Horne, Zeilinger; extension by Mermin |
| Field | Quantum mechanics / Quantum information |
Greenberger–Horne–Zeilinger state
The Greenberger–Horne–Zeilinger state (commonly abbreviated as GHZ state) is a class of entangled quantum states of three or more subsystems that exhibit strong multipartite correlations not reducible to pairwise entanglement. Introduced in the late 1980s by Greenberger, Horne, and Zeilinger, the GHZ state provides a striking demonstration of quantum nonlocality and plays a central role in studies of quantum entanglement, Bell's theorem, and applications in quantum information science.
The canonical three-qubit GHZ state is defined in the computational basis of three two-level systems (qubits) as |GHZ⟩ = (|000⟩ + |111⟩)/√2. More generally, an n-qubit GHZ state takes the form (|0...0⟩ + |1...1⟩)/√2. These pure states are elements of the tensor-product Hilbert space H = H_1 ⊗ H_2 ⊗ ... ⊗ H_n and are stabilized by operators from the Pauli group such as X^{⊗n} and correlated Z_i Z_j terms. The GHZ state is inequivalent under local operations and classical communication (LOCC) to the W state, illustrating different entanglement classes characterized by SLOCC invariants. Mathematical tools used to analyze GHZ states include density operators, reduced states, entanglement measures such as von Neumann entropy, and multipartite entanglement monotones.
GHZ states have been generated across multiple physical platforms. Early photonic realizations exploited spontaneous parametric down-conversion in nonlinear crystals combined with interferometric fusion pioneered in laboratories such as those led by Anton Zeilinger and groups at University of Innsbruck and University of Vienna. GHZ states have also been produced with trapped ions in experiments at NIST and Innsbruck using entangling gates, and in superconducting circuits at institutions including IBM and Google Quantum AI using tunable couplers. Other platforms include neutral atoms in optical lattices, nuclear magnetic resonance (NMR) systems, and solid-state defects such as NV centers in diamond. Experimental generation commonly employs entangling gates (e.g., controlled-NOT, Mølmer–Sørensen) or post-selected linear-optical schemes with path and polarization encoding.
GHZ states manifest genuine multipartite entanglement: no bipartition yields a separable state, as revealed by negative partial transpose tests on bipartitions and nonzero multipartite concurrence or tangle. The state exhibits maximal collective coherence between two macroscopically distinct components |0...0⟩ and |1...1⟩, leading to sensitivity in metrological applications characterized by the quantum Fisher information. Unlike the W state, GHZ entanglement is fragile under loss of a particle: tracing out a subsystem typically yields a mixed separable-like reduced state. Entanglement witnesses tailored to GHZ form and tomography techniques from quantum state tomography are used to certify multipartite entanglement in laboratory settings.
The GHZ paradox yields a deterministic contradiction between the predictions of local realism and quantum mechanics without invoking inequalities, strengthening the conclusions of Bell-type arguments. The original GHZ argument demonstrates that certain combinations of measurement outcomes on three separated qubits are incompatible with preassigned local hidden variables, emphasizing the nonclassicality of quantum correlations. GHZ states have been central to discussions by researchers such as John S. Bell, Niels Bohr, and modern experimental tests by groups at Orsay, Boulder, and Vienna. The paradox has influenced interpretations of quantum mechanics, debates on contextuality and inspired theoretical work on multipartite nonlocal games and Mermin's inequalities.
GHZ states are resources for protocols in quantum cryptography and distributed quantum tasks. They underpin schemes for quantum secret sharing, multipartite quantum teleportation, and conference key agreement. In quantum computing, GHZ states serve as testbeds for error detection codes and stabilizer-based fault-tolerant primitives, and they provide metrological advantages in protocols like quantum metrology and phase estimation achieving Heisenberg-limited scaling under ideal conditions. GHZ entanglement also appears in measurement-based quantum computation as cluster-state components and in demonstrations of quantum networks linking nodes at institutions such as Delft University of Technology and University of Science and Technology of China.
GHZ states are notably sensitive to decoherence channels such as amplitude damping, phase damping, and depolarizing noise. Environmental coupling in implementations at NIST, IBM, or optical laboratories leads to rapid degradation of off-diagonal coherence terms in the density matrix, reducing multipartite entanglement measures. Techniques to enhance robustness include entanglement purification, error-correcting codes (Shor, Steane), dynamical decoupling, and encoding logical GHZ states into decoherence-free subspaces. Studies in open quantum systems theory quantify lifetimes via master equations and experimentally evaluate fidelity versus noise models to optimize platform-specific control.
Generalizations of GHZ states include asymmetric superpositions, GHZ-diagonal mixed states, and high-dimensional GHZ states (qudits) of the form (|0...0⟩ + |d-1...d-1⟩)/√2. Cluster states, graph states, and stabilizer states relate closely to GHZ structure within the stabilizer formalism used in quantum error correction. Multipartite Bell inequalities and entanglement measures have been extended to characterize GHZ-type nonlocality for arbitrary n, and theoretical frameworks connect GHZ correlations to quantum networks, entanglement percolation, and resource theories of multipartite entanglement. Contemporary research at institutions like MIT, Caltech, and Max Planck Institute for Quantum Optics explores scalable generation, certification, and utilization of GHZ-like states in emerging quantum technologies.
Category:Quantum states Category:Quantum information theory