# Numerical Shadow

The web resource on numerical range and numerical shadow

### Site Tools

aux-definitions:ghz-state

# Differences

This shows you the differences between two versions of the page.

 aux-definitions:ghz-state [2013/03/11 09:15]lpawela created aux-definitions:ghz-state [2018/10/08 09:00] (current)plewandowska [Definition] 2018/10/08 09:00 plewandowska [Definition] 2018/05/22 14:56 plewandowska [Properties] 2014/02/04 02:09 lpawela [Properties] 2013/08/03 20:49 lpawela 2013/04/16 11:45 lpawela 2013/04/16 00:09 lpawela definitions:ghz-state renamed to aux-definitions:ghz-state2013/03/20 15:31 lpawela 2013/03/20 15:29 lpawela 2013/03/11 09:23 lpawela 2013/03/11 09:16 lpawela 2013/03/11 09:15 lpawela created Next revision Previous revision 2018/10/08 09:00 plewandowska [Definition] 2018/05/22 14:56 plewandowska [Properties] 2014/02/04 02:09 lpawela [Properties] 2013/08/03 20:49 lpawela 2013/04/16 11:45 lpawela 2013/04/16 00:09 lpawela definitions:ghz-state renamed to aux-definitions:ghz-state2013/03/20 15:31 lpawela 2013/03/20 15:29 lpawela 2013/03/11 09:23 lpawela 2013/03/11 09:16 lpawela 2013/03/11 09:15 lpawela created Line 1: Line 1: - 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​ [[qubits]] the '''​GHZ state'''​ can be written as + - :<​math>​|GHZ\rangle ​= \frac{|0\rangle^{\otimes ​M} + |1\rangle^{\otimes ​M}}{\sqrt{2}}.​ + ===== 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).​ + $$+ \ket{\mathrm{GHZ}} ​= \frac{1}{\sqrt{2}}\left( \ket{000}+\ket{111}\right). +$$ - == 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''​. Line 14: Line 21: 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|​ + $$- which is an unentagled [[mixed state]]. It has certain two-particle (qubit) correlations,​ but these are of a classical nature. + Tr_3\left((\ket{000}+\ket{111})(\bra{000}+\bra{111}) \right) = \ket{00}\bra{00} + \ket{11}\bra{11} - +$$ - 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​ or <​math>​|11\rangle​ 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. 