 Solution

If the eigenvalues of a density operator are $p_j\ge 0$ with ${\sum }_j{p}_j=1$, thenwith equality exactly when one eigenvalue is one, namely for a pure state. In dimension $d$, Cauchy--Schwarz gives $\gamma \ge 1/d$, with equality for the maximally mixed state. On an infinite-dimensional space the universal lower bound is zero as an infimum.