The determinant of a two-qubit (4 x 4) density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between 0 and (1/2)^8. (A "pure state" has determinant zero, and the fully mixed [classical] state, with 1/4's on its diagonal, determinant (1/2)^8.)

The determinant of the partial transpose (transpose in place the four 2 x 2 blocks) of such a matrix (nonnegative values indicating separability) lies between -(1/2)^4 and (1/2)^8. (The minimum is achieved by a "Bell state" and the maximum, again by the fully mixed state.)

What is the range (upper and lower limits) of the difference of these two determinants?

The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant zero, and the fully mixed [classical] state, with $\frac{1}{4}$'s on its diagonal, determinant $(\frac{1}{2})^8$.)

The determinant of the partial transpose (transpose in place the four $2 \times 2$ blocks) of such a matrix (nonnegative values indicating separability) lies between $-(\frac{1}{2})^4$ and $(\frac{1}{2})^8$. (The minimum is achieved by a "Bell state" and the maximum, again by the fully mixed state.)

What is the range (upper and lower limits) of the difference of these two determinants?