GHZ state

GHZ state. In quantum mechanics and quantum information theory, it is a specific type of entangled state involving three or more qubits. It is named for its proposers, physicists Daniel Greenberger, Michael Horne, and Anton Zeilinger, who introduced it to highlight profound contradictions with local hidden variable theories. The state exhibits maximal entanglement and serves as a crucial resource in protocols like quantum teleportation and quantum cryptography.

Definition and mathematical representation

The canonical form for three qubits is an equal superposition of two complementary computational basis states. In the Dirac notation common to quantum mechanics, it is written as (|000⟩ + |111⟩)/√2. This mathematical representation extends directly to systems with more particles, such as the N-qubit generalization (|0⟩^⊗N + |1⟩^⊗N)/√2. The state's structure is fundamentally linked to the Pauli matrices and their eigenstates, particularly the Z-basis. Preparation often involves sequences of Hadamard gate and controlled-NOT gate operations within a quantum circuit model, as described in textbooks like Quantum Computation and Quantum Information by Michael Nielsen and Isaac Chuang.

Properties and entanglement

It displays genuine multipartite entanglement, meaning the entanglement is shared among all particles and cannot be reduced to simpler Bell state correlations between pairs. A key property is its high sensitivity to particle loss; measuring or losing even one qubit collapses the entire state into a separable product state. This fragility, known as the "all-or-nothing" character, contrasts with the robustness of other states like the W state. The state also provides a stark violation of Bell's theorem through the GHZ theorem, offering a test of quantum nonlocality without statistical inequalities, unlike the CHSH inequality.

Experimental realization

The first experimental demonstrations were achieved in the late 1990s using systems like photons generated via spontaneous parametric down-conversion in nonlinear crystals such as beta barium borate. Pioneering work was conducted by groups including those at the University of Innsbruck and the National Institute of Standards and Technology. Later realizations utilized diverse platforms including trapped ions manipulated with laser pulses, superconducting qubits in devices from companies like IBM and Google, and nitrogen-vacancy centers in diamond. These experiments often verify entanglement through measurements of quantum state tomography or violations of Mermin's inequality.

Applications in quantum information

It serves as a critical resource in several advanced quantum protocols. In quantum communication, it enables schemes for quantum secret sharing and quantum Byzantine agreement, where information is split among multiple parties. Within quantum computing, it is used in certain quantum error correction codes and as a testbed for quantum gate fidelity in architectures like the surface code. The state is also central to foundational tests, such as closing the detection loophole in experiments at facilities like the Vienna Center for Quantum Science and Technology.

Relation to other quantum states

It represents one of two fundamental entanglement classes for three qubits under stochastic local operations and classical communication, the other being the W state. While it is equivalent to a GHZ-type state, it is distinct from cluster states used in measurement-based quantum computation. Its structure is a specific instance of a graph state corresponding to a star graph. The state is also connected to concepts in quantum thermodynamics and the study of quantum phase transitions in models like the transverse field Ising model studied at institutions like the Perimeter Institute for Theoretical Physics.

Category:Quantum information science Category:Quantum mechanics