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

Abstract A total set of n states | i 〉 and the corresponding projectors Π ( i ) = | i 〉 〈 i | are considered, in a quantum system with d -dimensional Hilbert space H ( d ) . A partially known density matrix ρ with given p ( i ) = Tr [ ρ Π ( i ) ] (where i = 1 , … , n and d ≤ n ≤ d 2 − 1 ) is considered, and its ranking permutation is defined. It is used to calculate the Choquet integral 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 C ( ρ ) ∕ Tr [ C ( ρ ) ] is a density matrix which is a good approximation to the partially known density matrix ρ .