on a Hilbert space $\mathcal{X}$. Let $\{\phi_1,\phi_2,\ldots\}$ and $\{\psi_1,\psi_2,\ldots\}$ be orthonormal bases of $\mathcal{X}$ consisting of eigenvectors of $A$ and $B$, respectively, with corresponding eigenvalues $p_i$ and $q_i$. We define $$\log A = \sum_{i: p_i > 0} \log(p_i) |\phi_i \rangle\!\langle \phi_i |\ ,$$ a negative-semidefinite, but generally unbounded, operator. Note that $\log A$ is defined so that $\ker(\log A)=\ker ( A )$. The von Neumann entropy of $A$ is defined to be $ S(A) = -\Tr ( A \log A ) = -\sum p_i \log(p_i)$. It may equal $+\infty$.

We define $-\Tr ( A \log B )$ to be $+\infty$ if $\ker B \not\subset \ker A$, otherwise, we define it by $$-\Tr ( A \log B ) \equals - \sum_i \sum_j | \langle \phi_i, \psi_j \rangle |^2 p_i \log q_j$$ as done in Ref. [@ArakiLieb]. We define the entropy of $A$ relative to $B$ by the formula $$S( A \| B ) \equals \sum_i \sum_j | \langle \phi_i, \psi_j \rangle |^2 ( p_i \log p_i - p_i \log q_j + q_j - p_i)$$ as done in Ref. [@Lindblad73]. The fact that the series defining $S( A \| B )$ has only nonnegative terms implies that $$\label{relative entropy}
      S( A \| 