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

Abstract

A total set of $n$ states $|i⟩$ and the corresponding projectors $\Pi (i)=|i⟩⟨i|$ are considered, in a quantum system with $d$-dimensional Hilbert space $H(d)$. A partially known density matrix $\rho$ with given $p(i)=\mathrm{Tr}[\rho \Pi (i)]$ (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 $\mathcal{C}(\rho )$ 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 $\mathcal{C}(\rho )/\mathrm{Tr}[\mathcal{C}(\rho )]$ is a density matrix which is a good approximation to the partially known density matrix $\rho$.

References in corpus (1)

• Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces

Cited by in corpus (2)

• Uncertainty relations in terms of the Gini index for finite quantum systems
• Equivalence classes of coherent projectors in a Hilbert space with prime dimension: Q functions and their Gini index