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 (particles). It was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989.^[1] Extremely non-classical properties of the state have been observed.

In the case of each of the subsystems being two-dimensional, that is for qubits, 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 GHZ state is a maximally entangled quantum state.

The simplest one is the 3-qubit GHZ state:

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.

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.

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). 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 naive measurement of the third particle of the GHZ state results in an unentangled pair, a more clever measurement, along an orthogonal direction, can leave behind a maximally entangled Bell state. This is illustrated below. The lesson to be drawn from this is that pairwise entanglement in the GHZ is more subtle than it naively appears: measurements along the privileged Z direction destroy pairwise entanglement, but other measurements (along different axes) do not.

The GHZ state can be written as

The point of this example is that it illustrates that the pairwise entanglement of the GHZ state is more subtle than it first appears: a judicious measurement along an orthogonal direction, along with the application of a quantum transform depending 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.^[4]