GHZ Theorem And Wigner-Yanase Entanglement

Quantum entanglement is a special quantum mechanical resource that plays a keyrole in many applications of quantum computation and quantum information. Con-sequently, it is significant to study quantitatively and qualitatively its application inthe field of quantum information theory. In this thesis, we focus our research inGreenberger-Home-Zeilinger (GHZ) theorem for the four-qubit system, and the ap-plication of a measure of entanglement in terms of Wigner-Yanase information:â… . GHZ theorem constitutes that there is a set of mutually commuting nonlocalobservables with a common eigenstate on which those observables assume values thatrefute the attempt to assign values only required to have them by the local realismof Einstein, Podolsky, and Rosen (EPR). It is known that there is only one formof the GHZ-Mermin-like argument with equivalence up to a local unitary transfor-marion, which is exactly Mermin's version of the GHZ theorem. However, for afour-qubit system which was originally studied by GHZ, we show that there are 9 dis-tinct forms of the GHZ-Mermin-like argument. The proof is obtained by using somegeometric invariants to characterize the sets of mutually commuting nonlocal spin ob-servables 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 observablesin the four-qubit system, and each GHZ-Mermin-like argument involves a set of atleast five mutually commuting four-qubit nonlocal spin observables with a GHZ stateas a common eigenstate in GHZ's theorem. Therefore, we present a complete con-struction of the GHZ theorem for the four-qubit system.â…¡. A measure of entanglement on n qubits is defined in terms of Wigner-Yanaseinformation. It is shown that the measure coincides essentially with the concurrenceon two qubits. This uncovers the information-theoretic meaning of the concurrence ofentangled states.