Complete sets of commuting observables of Greenberger-Horne-Zeilinger states

Complete sets of commuting observables (CSCOs) of the form Sigma(N)=Pi(i=1)(N)sigma(ialphai) (alpha(i)=x,y,z) for an N-qubit system are extracted by a simple graphic approach. One can construct 2x3(N) sets of operators, each set consisting of K-N commuting Sigma(N), K-N=2(N-1)+1 for even N, and 2(N-1) for odd N. Any N functional-independent operators among the K-N 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 (Ngreater than or equal to3) systems. Sigma(N) may be considered as the infinitesimal operator of rotational operator R(alpha(1),alpha(2),...alpha(N))=Pi(i=1)(N)exp[-ipisigma(ialphai)/2] , whose eigenvalue (signature) r=e(-ipialpha), or signature exponent alpha, may be equivalently used for characterizing each basis.