As the simultaneous eigenstates of a complete set of commuting observables (CSCO) consisting of threebody operators of the form σ1ασ2βσrsγ, the construction of entangled states for a three-particle system is investigated. It is shown that there exist 54 different sets of operators (each containing four commuting three-body operators with their product being -1) and any three members of each set constitute a CSCO. It is found that 54 different sets of maximally entangled states can be constructed, including the usual GHZ (Greenberg-Horne-Zeilinger) state as a special case. Once a preferential representation is adopted, all of them can be expressed as a GHZ-like form. Moreover, there also exist 18sets of partially entangled states, where only two particles are entangled.
As the simultaneous eigenstates of a complete set of commuting observables (CSCO) consisting of three-body operators of the form , the construction of entangled states for a three-particle system is investigated. It is shown that there exist 54 different sets of operators (each containing four commuting three-body operators with their product being -1) and any three members of each set constitute a CSCO. It is found that 54 different sets of maximally entangled states can be constructed, including the usual GHZ (Greenberg-Horne-Zeilinger) state as a special case. Once a preferential representation is adopted, all of them can be expressed as a GHZ-like form. Moreover, there also exist 18 sets of partially entangled states, where only two particles are entangled.
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.
Complete sets of commuting observables (CSCOs) of the form ${\ensuremath{\Sigma}}_{N}={\ensuremath{\Pi}}_{i=1}^{N}{\ensuremath{\sigma}}_{i{\ensuremath{\alpha}}_{i}}\phantom{\rule{0.3em}{0ex}}({\ensuremath{\alpha}}_{i}=x,y,z)$ for an
N-qubit system are extracted by a simple graphic approach. One can construct $2\ifmmode\times\else\texttimes\fi{}{3}^{N}$ sets of operators, each set consisting of
KN commuting ${\ensuremath{\Sigma}}_{N}$, ${K}_{N}={2}^{N\ensuremath{-}1}+1$ for even
N, and ${2}^{N\ensuremath{-}1}$ for odd
N. Any
N functional-independent operators among the
KN operators may be adopted as a CSCO, whose simultaneous eigenstates (SEs) span an orthonormal basis of
N-qubit space. These SEs have reduced density matrix of rank 2 and can be reduced to the Greenberger-Horne-Zeilinger (GHZ) state form of Eq. (2) in suitable representations. The all-versus-nothing demolition of the elements of reality holds for each basis of the form of Eq. (2) for
N-qubit $(N\ensuremath{\geqslant}3)$ systems. ${\ensuremath{\Sigma}}_{N}$ may be considered as the infinitesimal operator of rotational operator $R({\ensuremath{\alpha}}_{1},{\ensuremath{\alpha}}_{2},\dots{}{\ensuremath{\alpha}}_{N})={\ensuremath{\Pi}}_{i=1}^{N}\mathrm{exp}[\ensuremath{-}i\ensuremath{\pi}{\ensuremath{\sigma}}_{i{\ensuremath{\alpha}}_{i}}∕2]$ , whose eigenvalue (signature) $r={e}^{\ensuremath{-}i\ensuremath{\pi}\ensuremath{\alpha}}$, or signature exponent $\ensuremath{\alpha}$, may be equivalently used for characterizing each basis.
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable,
W, and Greenberger-Horne-Zeilinger (GHZ) classes. The
W and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the
W and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.