In this paper, some general results about the concepts of complete sets of commuting observables (CSCO) and irreducible sets of observables (ISO) are obtained. It is proved the following: (i) Any relevant observable is an essential part of some CSCO. (ii) Any relevant observable, which is a CSCO, is an essential part of some ISO. (iii) Let {b α } α ∈ I be a CSCO, or an ISO, then, I is countable
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 self adjoint linear operators which represent the observables of a physical system are in general not an irreducible system. Because a complete set of commuting observables must determine the state of a physical system unambiguously the observables generate an algebra of operators which must contain a maximal abelian subalgebra. The structure of such algebras is investigated and it is shown by applying the theory of the direct integral of Hilbert spaces that there exists always a unique canonical representation of the Hilbert space as a direct integral in such a way that the transformations which are induced by the observables in the component subspaces are irreducible.
From here, we start relaxing the constraints that we had imposed in the earlier chapters for simplicity, and start moving towards more general scenarios. First, measurements with inadequate resolution, employing insufficiently selective measurement devices have been introduced. In this book they have been termed imprecise measurements. Here, we admit for the first time that all measurements do not prepare quantum states. We argue how the description of imprecise observables requires the use of Hermitian operators with degenerate eigenvalues: degeneracy encodes insufficient selectiveness. The general law of state preparation, the standard jargon for which is “collapse postulate”, is described in its full glory here. We then move on to introduce the idea of compatibility of observables and demonstrate how certain groups of imprecise measurements can collaborate to prepare states. This leads to the important idea of “complete set of compatible observables” and the related concepts of “complete and incomplete measurements”. Subsequently, the association of compatible observables and commuting operators is established which also connects complete set of compatible observables and complete set of commuting operators (both of which are accronymed CSCO, without ambiguity). We also give an explicit illustration of the algebraic construction of a CSCO.