aux-definitions:ghz-state

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aux-definitions:ghz-state [2013/03/11 09:15] lpawela created |
aux-definitions:ghz-state [2018/10/08 09:00] (current) plewandowska [Definition] |
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- | A '''Greenberger-Horne-Zeilinger''' state is an [[Theory of entanglement|entangled]] [[states|quantum state]] having extremely non-classical properties. | + | ====== GHZ state ====== |

- | == Definition == | + | A '''Greenberger-Horne-Zeilinger''' state is an entangled quantum state having extremely non-classical properties. |

- | For a system of <math>n</math> [[qubits]] the '''GHZ state''' can be written as | + | |

- | :<math>|GHZ\rangle = \frac{|0\rangle^{\otimes M} + |1\rangle^{\otimes M}}{\sqrt{2}}.</math> | + | ===== 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: | The simplest one is the 3-qubit GHZ state is: | ||

- | :<math>|GHZ\rangle = \frac{1}{\sqrt{2}}\left( |000\rangle+|111\rangle\right).</math> | + | $$ |

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

+ | $$ | ||

- | == Properties == | + | ===== Properties ===== |

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

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

we get | we get | ||

- | :<math>Tr_3\left((|000\rangle + |111\rangle)(\langle 000|+\langle 111|) \right) = |00\rangle \langle 00| + |11\rangle \langle 11|</math> | + | $$ |

- | which is an unentagled [[mixed state]]. It has certain two-particle (qubit) correlations, but these are of a classical nature. | + | Tr_3\left((\ket{000}+\ket{111})(\bra{000}+\bra{111}) \right) = \ket{00}\bra{00} + \ket{11}\bra{11} |

- | | + | $$ |

- | 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 <math>|00\rangle</math> or <math>|11\rangle</math> which are unentangled pure states. This is unlike the [[W state]] which leaves bipartite entanglements even when we measure one of its subsystems. | + | 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). They can be easily shown to invalidate the ideas of Einstein (see [[EPR Paradox]]). This is an amplification of the [[Bell's theorem]]. The correlations can be utilized in some [[quantum information]] tasks. These include multipartner [[quantum cryptography]] (1998) and [[communication complexity]] tasks (1997, 2004). | + | |

- | | + | |

- | === See also === | + | |

- | * Daniel M. Greenberger, Michael A. Horne, Abner Shimony, Anton Zeilinger, Bell's theorem without inequalities, Am. J. Phys. 58 (12), 1131 (1990); | + | |

- | | + | |

- | {{stub}} | + | |

- | [[Category:Quantum States]] | + | 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 [[aux-definitions:w-state|W state]] which leaves bipartite entanglements even when we measure one of its subsystems. |

aux-definitions/ghz-state.1362993345.txt.gz · Last modified: 2013/03/11 09:15 by lpawela