aux-definitions:ghz-state

A `'Greenberger-Horne-Zeilinger`

' state is an entangled quantum state having extremely non-classical properties.

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

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.

aux-definitions/ghz-state.txt · Last modified: 2018/10/08 09:00 by plewandowska