dorsal/arxiv
View SchemaComprehensive proof of the Greenberger-Horne-Zeilinger Theorem for the four-qubit system
| Authors | Li Tang, Jie Zhong, Yaofeng Ren, Mingsheng Zhan, Zeqian Chen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0509158 |
| URL | https://arxiv.org/abs/quant-ph/0509158 |
Abstract
Greenberger-Horne-Zeilinger (GHZ) theorem asserts that there is a set of mutually commuting nonlocal observables with a common eigenstate on which those observables assume values that refute the attempt to assign values only required to have them by the local realism of Einstein, Podolsky, and Rosen (EPR). It is known that for a three-qubit system there is only one form of the GHZ-Mermin-like argument with equivalence up to a local unitary transformation, which is exactly Mermin's version of the GHZ theorem. In this paper, however, for a four-qubit system which was originally studied by GHZ, we show that there are nine distinct forms of the GHZ-Mermin-like argument. The proof is obtained by using some geometric invariants to characterize the sets of mutually commuting nonlocal spin observables on the four-qubit system. It is proved that there are at most nine elements (except for a different sign) in a set of mutually commuting nonlocal spin observables in the four-qubit system, and each GHZ-Mermin-like argument involves a set of at least five mutually commuting four-qubit nonlocal spin observables with a GHZ state as a common eigenstate in GHZ's theorem. Therefore, we present a complete construction of the GHZ theorem for the four-qubit system.
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"abstract": "Greenberger-Horne-Zeilinger (GHZ) theorem asserts that there is a set of\nmutually commuting nonlocal observables with a common eigenstate on which those\nobservables assume values that refute the attempt to assign values only\nrequired to have them by the local realism of Einstein, Podolsky, and Rosen\n(EPR). It is known that for a three-qubit system there is only one form of the\nGHZ-Mermin-like argument with equivalence up to a local unitary transformation,\nwhich is exactly Mermin\u0027s version of the GHZ theorem. In this paper, however,\nfor a four-qubit system which was originally studied by GHZ, we show that there\nare nine distinct forms of the GHZ-Mermin-like argument. The proof is obtained\nby using some geometric invariants to characterize the sets of mutually\ncommuting nonlocal spin observables on the four-qubit system. It is proved that\nthere are at most nine elements (except for a different sign) in a set of\nmutually commuting nonlocal spin observables in the four-qubit system, and each\nGHZ-Mermin-like argument involves a set of at least five mutually commuting\nfour-qubit nonlocal spin observables with a GHZ state as a common eigenstate in\nGHZ\u0027s theorem. Therefore, we present a complete construction of the GHZ theorem\nfor the four-qubit system.",
"arxiv_id": "quant-ph/0509158",
"authors": [
"Li Tang",
"Jie Zhong",
"Yaofeng Ren",
"Mingsheng Zhan",
"Zeqian Chen"
],
"categories": [
"quant-ph"
],
"title": "Comprehensive proof of the Greenberger-Horne-Zeilinger Theorem for the four-qubit system",
"url": "https://arxiv.org/abs/quant-ph/0509158"
},
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