Dichotomic probability representation of quantum states

We present systematic proofs of statements about probability representations of qu-Nit density states, which are Hermitian trace-one nonnegative matrices; these properties provide new relations and new entropic-information inequalities for the matrices. We give a rigorous proof of the expression of mentioned NxN matrices in terms of standard probability distributions of dichotomic random variables. The examples of qu-trit and qu-quart states are explicitly considered, and new properties of roots of the characteristic polynomial associated with trace-one nonnegative Hermitian matrices are provided.