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

Abstract

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 $d\le n\le d^2-1$) is considered, and its ranking permutation is defined. It is used to calculate the Choquet integral ${\cal C}(\rho)$ 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 ${\cal C}(\rho)/{\rm Tr}[{\cal C}(\rho)]$ is a density matrix which is a good approximation to the partially known density matrix $\rho$.