One of the aims of information theory is to find out the capacity of noisy channels defined as the ultimate limit on communication rates. If an optical communication link is modeled quantum mechanically, then the task of finding the capacity (for sending classical data) relies on certain minimum output entropy conjecture which in turn is implied by the entropy photon number inequality (EPnI) conjecture. It has been a longstanding problem in quantum information theory to resolve these conjectures. We show that the 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 (joint work with Smarajit Das and Siddharth Muthukrishnan).