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

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

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