GHZ state

A Greenberger-Horne-Zeilinger state is an entangled quantum state having extremely non-classical properties.

Definition

For a system of $ d $ qubits the GHZ state can be written as

\[\ket{\mathrm{GHZ}} = \frac{\ket{0}^{\otimes d} + \ket{1}^{\otimes d}}{\sqrt{2}}.\]

The simplest one is the 3-qubit GHZ state is:

\[\ket{\mathrm{GHZ}} = \frac{1}{\sqrt{2}}\left( \ket{000}+\ket{111}\right).\]

Properties

Apparently there is no standard measure of multi-partite entanglement, but many measures define the GHZ to be maximally entangled.

Important property of the GHZ state is that when we trace over one of the three systems we get

\[\Tr_3\left((\ket{000}+\ket{111})(\bra{000}+\bra{111}) \right) = \ket{00}\bra{00} + \ket{11}\bra{11}\]

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 subsystems, in such a way that the measurement distinguishes between the states 0 and 1, we will leave behind either $\ket{00}$ or $\ket{11}$ which are unentangled pure states. This is unlike the W state which leaves bipartite entanglements even when we measure one of its subsystems.