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A total set of $n$ states $|i\rangle$ and the corresponding projectors$\Pi(i)=|i\rangle \langle i|$ are considered, in a quantum system with$d$-dimensional Hilbert space $H(d)$. A partially known density matrix $\rho$with given $p(i)={\rm Tr}[\rho \Pi(i)]$ (where $i=1,...,n$ and $d\le n\led^2-1$) is considered, and its ranking permutation is defined. It is used tocalculate the Choquet integral ${\cal C}(\rho)$ which is a positivesemi-definite Hermitian matrix. Comonotonicity is an important concept in theformalism, which is used to formalise the vague concept of physically similardensity matrices. It is shown that ${\cal C}(\rho)/{\rm Tr}[{\cal C}(\rho)]$ isa density matrix which is a good approximation to the partially known densitymatrix $\rho$.

The Choquet integral as an approximation to density matrices with incomplete information