In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state (GHZ state) is a certain type of entangled quantum state that involves at least three subsystems (particle states, qubits, or qudits). The four-particle version was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989, and the three-particle version was introduced by N. David Mermin in 1990. [1] [2] [3] Extremely non-classical properties of the state have been observed. GHZ states for large numbers of qubits are theorized to give enhanced performance for metrology compared to other qubit superposition states. [4]
The GHZ state is an entangled quantum state for 3 qubits and its state is
The generalized GHZ state is an entangled quantum state of M > 2 subsystems. If each system has dimension , i.e., the local Hilbert space is isomorphic to , then the total Hilbert space of an -partite system is . This GHZ state is also called an -partite qudit GHZ state. Its formula as a tensor product is
In the case of each of the subsystems being two-dimensional, that is for a collection of M qubits, it reads
There is no standard measure of multi-partite entanglement because different, not mutually convertible, types of multi-partite entanglement exist. Nonetheless, many measures define the GHZ state to be maximally entangled state.[ citation needed]
Another important property of the GHZ state is that taking the partial trace over one of the three systems yields
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.[ citation needed]
The GHZ state is non-biseparable [5] and is the representative of one of the two non-biseparable classes of 3-qubit states which cannot be transformed (not even probabilistically) into each other by local quantum operations, the other being the W state, . [6] Thus and represent two very different kinds of entanglement for three or more particles. [7] The W state is, in a certain sense "less entangled" than the GHZ state; however, that entanglement is, in a sense, more robust against single-particle measurements, in that, for an N-qubit W state, an entangled (N − 1)-qubit state remains after a single-particle measurement. By contrast, certain measurements on the GHZ state collapse it into a mixture or a pure state.
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 article. The first laboratory observation of GHZ correlations was by the group of Anton Zeilinger (1998), who was awarded a share of the 2022 Nobel Prize in physics for this work. [8] 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).
Although a measurement of the third particle of the GHZ state that distinguishes the two states results in an unentangled pair, a measurement along an orthogonal direction can leave behind a maximally entangled Bell state. This is illustrated below.
The 3-qubit GHZ state can be written as
where the third particle is written as a superposition in the X basis (as opposed to the Z basis) as and .
A measurement of the GHZ state along the X basis for the third particle then yields either , if was measured, or , if was measured. In the later case, the phase can be rotated by applying a Z quantum gate to give , while in the former case, no additional transformations are applied. In either case, the result of the operations is a maximally entangled Bell state.
This example illustrates that, depending on which measurement is made of the GHZ state is more subtle than it first appears: a measurement along an orthogonal direction, followed by a quantum transform that depends on the measurement outcome, can leave behind a maximally entangled state.
GHZ states are used in several protocols in quantum communication and cryptography, for example, in secret sharing [9] or in the quantum Byzantine agreement.
Mermin was the first to point out the interesting properties of this three-system state, following the lead of D. M. Greenberger, M. Horne, and A. Zeilinger [...] where a similar four-system state was proposed.
In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state (GHZ state) is a certain type of entangled quantum state that involves at least three subsystems (particle states, qubits, or qudits). The four-particle version was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989, and the three-particle version was introduced by N. David Mermin in 1990. [1] [2] [3] Extremely non-classical properties of the state have been observed. GHZ states for large numbers of qubits are theorized to give enhanced performance for metrology compared to other qubit superposition states. [4]
The GHZ state is an entangled quantum state for 3 qubits and its state is
The generalized GHZ state is an entangled quantum state of M > 2 subsystems. If each system has dimension , i.e., the local Hilbert space is isomorphic to , then the total Hilbert space of an -partite system is . This GHZ state is also called an -partite qudit GHZ state. Its formula as a tensor product is
In the case of each of the subsystems being two-dimensional, that is for a collection of M qubits, it reads
There is no standard measure of multi-partite entanglement because different, not mutually convertible, types of multi-partite entanglement exist. Nonetheless, many measures define the GHZ state to be maximally entangled state.[ citation needed]
Another important property of the GHZ state is that taking the partial trace over one of the three systems yields
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.[ citation needed]
The GHZ state is non-biseparable [5] and is the representative of one of the two non-biseparable classes of 3-qubit states which cannot be transformed (not even probabilistically) into each other by local quantum operations, the other being the W state, . [6] Thus and represent two very different kinds of entanglement for three or more particles. [7] The W state is, in a certain sense "less entangled" than the GHZ state; however, that entanglement is, in a sense, more robust against single-particle measurements, in that, for an N-qubit W state, an entangled (N − 1)-qubit state remains after a single-particle measurement. By contrast, certain measurements on the GHZ state collapse it into a mixture or a pure state.
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 article. The first laboratory observation of GHZ correlations was by the group of Anton Zeilinger (1998), who was awarded a share of the 2022 Nobel Prize in physics for this work. [8] 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).
Although a measurement of the third particle of the GHZ state that distinguishes the two states results in an unentangled pair, a measurement along an orthogonal direction can leave behind a maximally entangled Bell state. This is illustrated below.
The 3-qubit GHZ state can be written as
where the third particle is written as a superposition in the X basis (as opposed to the Z basis) as and .
A measurement of the GHZ state along the X basis for the third particle then yields either , if was measured, or , if was measured. In the later case, the phase can be rotated by applying a Z quantum gate to give , while in the former case, no additional transformations are applied. In either case, the result of the operations is a maximally entangled Bell state.
This example illustrates that, depending on which measurement is made of the GHZ state is more subtle than it first appears: a measurement along an orthogonal direction, followed by a quantum transform that depends on the measurement outcome, can leave behind a maximally entangled state.
GHZ states are used in several protocols in quantum communication and cryptography, for example, in secret sharing [9] or in the quantum Byzantine agreement.
Mermin was the first to point out the interesting properties of this three-system state, following the lead of D. M. Greenberger, M. Horne, and A. Zeilinger [...] where a similar four-system state was proposed.