Many properties of entropic quantities are shown by resorting to related properties of the relative entropy of suitable quantities. For instance, subadditivity of entropy may follow from non negativity of relative entropy by observing that
$$H(A)_{\rho^{A}}+H(B)_{\rho^{B}}-H(A, B)_{\rho^{AB}}=D\left(\rho^{AB} \vert\vert \rho^A\otimes \rho^{B}\right)\,,$$
or concavity of conditional entropy from convexity of relative entropy by observing that
$$H(A\vert B)_{\rho^{AB}}=-D\left(\rho^{AB} \vert\vert I_A\otimes \rho^{B}\right)\,.$$
To obtain such identities, the following formula
$$\log\left( P\otimes Q\right)= \log P\otimes I_B+I_A\otimes \log Q$$
is often invoked. Here I assume the logarithm of a positive definite operator to be defined by means of the spectral decomposition; the eignevalues of a positive definite operator are all positive, so their logarithms (as positive real numbers) make sense. Essentially, if $P=\sum_p p \vert p\rangle$, then $\log P=\sum_p \log p \vert p\rangle$.
However, what happens when $P$ and $Q$ are positive semidefinite but not necessarily definite? This is especially relevant in the context of entropic quantities, where a density matrix may have the zero eigenvalue. Each book I have consulted makes [possibly – see this comment for some "refs"] use of $\log \left( P\otimes Q\right)= \log P\otimes I+I\otimes \log Q$ . How do we justify the use of such formula for $P$ and $Q$ semidefinite?
PS: In the semidefinite case I'd expect the identity operator to be replaced by a projection on the support, i.e., $\log \left( P\otimes Q\right)= \log P\otimes \Pi_Q+\Pi_P\otimes \log Q$. On the other hand, I can't see different well reputed textbooks to be wrong on the same point. Should, by any chance, my guess make sense, there would be the problem of deriving expressions such as $H(A: B)=D\left(\rho^{AB} \vert\vert \rho^A\otimes \rho^{B}\right)$, or $H(A\vert B)=-D\left(\rho^{AB} \vert\vert I_A\otimes \rho^{B}\right)$. Is there some trick when we take partial traces?
