Energy entropy relation in extremal density matrices for qudit states

The extremal density matrices are defined as the ones that make the mean value of the energy an extremal point. Using the Bloch vector factorization, a general procedure to calculate the extremal density matrices for any qudit system is presented. As examples, the general result for d=2 and some examples for d=3 and d=4 are obtained. The energy-entropy relation is discussed for these type of density matrices