Lemma: Entropy comparisons

For finite-dimensional states:

1. $S(\rho )\le H(\mathrm{diag}\rho )$ in any orthonormal basis;

2. $S$ is concave;

3. $S(\rho \otimes \sigma )=S(\rho )+S(\sigma )$;

4. a two-by-two state of determinant $u$ has entropy $g_2(u)$.

Proof. The diagonal is obtained from the eigenvalue vector by the doubly stochastic matrix of squared moduli of unitary entries. Concavity of $-x{\log }_{2}x$, applied row by row, proves item 1.

For a mixture, choose a basis diagonalizing that mixture. Classical entropy concavity in this basis, followed by item 1 for each component, proves item 2. The eigenvalues of a tensor product are pairwise products, which proves item 3. For item 4, the characteristic polynomial is ${\lambda }^2-\lambda +u$, giving the two eigenvalues used in (eq:g).