I have classes of $4 \times 4$ Hermitian matrices $D$, the two $2 \times 2$ diagonal blocks of which are themselves diagonal. That is, the (1,2),(2,1),(3,4) and (4,3) entries of $D$ are zero. (The other off-diagonal entries of $D$ can be real, complex or quaternionic.) I want necessary and sufficient (maybe minimal) conditions that $D$ be positive definite, that do not involve the full determinant of $D$. (In the quaternionic case, this would be the "Moore determinant", but the real and complex cases are the ones of immediate interest to me.)
As some background, this pertains to a quantum-information-theoretic problem, in which I am also interested in the "partial transpose" $D^{PT}$ of $D$, obtained, in general, by transposing in place the four $2 \times 2$ blocks of $D$. The determinants of both $D$ and $D^{PT}$ are quite cumbersome, while the difference of the two determinants simplifies greatly, which is the basic motivation for my question. (I want to enforce the positive-definite nature of $D^{PT}$, subject to $D$ itself being positive-definite. Hopefully, the $2 \times 2$ and $3 \times 3$ minors of these matrices are more computationally amenable than their determinants. This pertains to the problem of determining the probability that two quantum bits ["qubits"] are disentangled/separable.)