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

Abstract

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 ${K}_{N}$ 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 ${K}_{N}$ 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 $(N\ensuremath{\geqslant}3)$ 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.