Greenberger-Horne-Zeilinger state
Encyclopedia
In physics
, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled
quantum state which involves at least three subsystems (particles). It was first studied by D. Greenberger, M.A. Horne and Anton Zeilinger
in 1989. They have noticed the extremely non-classical properties of the state.
quantum state of M>2 subsystems. In the case of each of the subsystems being two-dimensional, that is for qubit
s, it reads
In simple words it is a quantum superposition of all subsystems being in state 0 with all of them being in state 1 (states 0 and 1 of a single subsystem are fully distinguishable).
The simplest one is the 3-qubit GHZ state:
Another important property of the GHZ state is that when we trace over one of the three systems
we get
which is an unentangled mixed state. It has certain two-particle (qubit) correlations, but these are of a classical nature.
On the other hand, if we were to measure one of the subsystems, in such a way that the measurement distinguishes between the states 0 and 1, we will leave behind either
or 
which are unentangled pure states. This is unlike the W state
which leaves bipartite entanglements even when we measure one of its subsystems.
The GHZ state leads to striking non-classical correlations (1989). Particles prepared in this state lead to a version of Bell's theorem
, which shows the internal inconsistency of the notion of elements-of-reality introduced in the famous Einstein–Podolsky–Rosen paper. The first laboratory observation of GHZ correlations was by the group of Anton Zeilinger
(1998). Many, more accurate observations followed. The correlations can be utilized in some quantum information
tasks. These include multipartner quantum cryptography
(1998) and communication complexity
tasks (1997, 2004).
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
quantum state which involves at least three subsystems (particles). It was first studied by D. Greenberger, M.A. Horne and Anton Zeilinger
Anton Zeilinger
Anton Zeilinger is an Austrian quantum physicist. He is currently professor of physics at the University of Vienna, previously University of Innsbruck. He is also the director of the Vienna branch of the Institute for Quantum Optics and Quantum Information IQOQI at the Austrian Academy of Sciences...
in 1989. They have noticed the extremely non-classical properties of the state.
Definition
The GHZ state is an entangledQuantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
quantum state of M>2 subsystems. In the case of each of the subsystems being two-dimensional, that is for qubit
Qubit
In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....
s, it reads
In simple words it is a quantum superposition of all subsystems being in state 0 with all of them being in state 1 (states 0 and 1 of a single subsystem are fully distinguishable).
The simplest one is the 3-qubit GHZ state:
Properties
Apparently there is no standard measure of multi-partite entanglement, but many measures define the GHZ to be maximally entangled.Another important property of the GHZ state is that when we trace over one of the three systems
we get
which is an unentangled mixed state. It has certain two-particle (qubit) correlations, but these are of a classical nature.
On the other hand, if we were to measure one of the subsystems, in such a way that the measurement distinguishes between the states 0 and 1, we will leave behind either
or which are unentangled pure states. This is unlike the W stateW state
The W state is an entangled quantum state of three qubits which has the following shapeand which is remarkable for representing a specific type of multipartite entanglement and for occurring in several applications in quantum information theory.-Properties:...
which leaves bipartite entanglements even when we measure one of its subsystems.
The GHZ state leads to striking non-classical correlations (1989). Particles prepared in this state lead to a version of Bell's theorem
Bell's theorem
In theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
, which shows the internal inconsistency of the notion of elements-of-reality introduced in the famous Einstein–Podolsky–Rosen paper. The first laboratory observation of GHZ correlations was by the group of Anton Zeilinger
Anton Zeilinger
Anton Zeilinger is an Austrian quantum physicist. He is currently professor of physics at the University of Vienna, previously University of Innsbruck. He is also the director of the Vienna branch of the Institute for Quantum Optics and Quantum Information IQOQI at the Austrian Academy of Sciences...
(1998). Many, more accurate observations followed. The correlations can be utilized in some quantum information
Quantum information
In quantum mechanics, quantum information is physical information that is held in the "state" of a quantum system. The most popular unit of quantum information is the qubit, a two-level quantum system...
tasks. These include multipartner quantum cryptography
Quantum cryptography
Quantum key distribution uses quantum mechanics to guarantee secure communication. It enables two parties to produce a shared random secret key known only to them, which can then be used to encrypt and decrypt messages...
(1998) and communication complexity
Communication complexity
The notion of communication complexity was introduced by Yao in 1979,who investigated the following problem involving two separated parties . Alice receives an n-bit string x and Bob another n-bit string y, and the goal is for one of them to compute a certain function f with the least amount of...
tasks (1997, 2004).
See also
- Bell's theoremBell's theoremIn theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
- Bell stateBell stateThe Bell states are a concept in quantum information science and represent the simplest possible examples of entanglement. They are named after John S. Bell, as they are the subject of his famous Bell inequality. An EPR pair is a pair of qubits which jointly are in a Bell state, that is, entangled...
- GHZ experimentGHZ experimentGHZ experiments are a class of physics experiments that may be used to generate starkly contrasting predictions from local hidden variable theory and quantum mechanical theory, and permit immediate comparison with actual experimental results. A GHZ experiment is similar to a test of Bell's...
- Local hidden variable theoryLocal hidden variable theoryIn quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have no instantaneous effect on local ones....
- Quantum entanglementQuantum entanglementQuantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
- QubitQubitIn quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....
- Measurement in quantum mechanicsMeasurement in quantum mechanicsThe framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus....