On some special cases of the Entropy Photon-Number Inequality

We show that the Entropy Photon-Number Inequality (EPnI) holds where one of the input states is the vacuum state and for several candidates of the other input state that includes the cases when the state has the eigenvectors as the number states and either has only two non-zero eigenvalues or has arbitrary number of non-zero eigenvalues but is a high entropy state. We also discuss the conditions, which if satisfied, would lead to an extension of these results.