Equivalent determinant condition for Peres-Horodecki criteria

Question

The Peres-Horodecki criteria for a two-qubit state states that if the smallest eigenvalue of the partial transpose of the state is negative, it is entangled, else it is separable. According to this paper (arXiv) page 4, left side, the following is an equivalent formulation to express the above criterion: Assume the matrix in question looks like $$\begin{bmatrix} \rho_{00,00} & \rho_{00,01} & \rho_{00,10} & \rho_{00,11} \\ \rho_{01,00} & \rho_{01,01} & \rho_{01,10} & \rho_{01,11} \\ \rho_{10,00} & \rho_{10,01} & \rho_{10,10} & \rho_{10,11} \\ \rho_{11,00} & \rho_{11,01} & \rho_{11,10} & \rho_{11,11} \\ \end{bmatrix}$$ Consider the following three determinants: \begin{align*} W_2 &= \begin{vmatrix} \rho_{00,00} & \rho_{01,00} \\ \rho_{00,01} & \rho_{01,01} \\ \end{vmatrix} \\ W_3 &= \begin{vmatrix} \rho_{00,00} & \rho_{01,00} & \rho_{00,10} \\ \rho_{00,01} & \rho_{01,01} & \rho_{00,11} \\ \rho_{10,00} & \rho_{11,00} & \rho_{10,10} \\ \end{vmatrix}\\ W_4 &= \begin{vmatrix} \rho_{00,00} & \rho_{01,00} & \rho_{00,10} & \rho_{01,10} \\ \rho_{00,01} & \rho_{01,01} & \rho_{00,11} & \rho_{01,11}\\ \rho_{10,00} & \rho_{11,00} & \rho_{10,10} & \rho_{11,10}\\ \rho_{10,01} & \rho_{11,01} & \rho_{10,11} & \rho_{11,11}\\ \end{vmatrix} \end{align*} Notice that $W_4$ is the determinant of the partial transpose of the matrix and $W_3$ and $W_2$ are the first $3\times 3$ and $2\times 2$ elements of the partial transpose. The condition is if $W_2 \geq 0$ and ($W_3 < 0$ or $W_4 < 0$), then the state is entangled. If not, it is separable. How are these two equivalent? Also, can this method be extended to ensure the smallest eigenvalue is greater than any $x$, where $x$ is not necessarily 0?

Answer 1

This is called Sylvester's Criterion. There's plenty of information available once you have the name. The linked wikipedia article contains a proof.

Strictly, Sylvester's Criterion requires that $W_2,W_3,W_4> 0$ for the state to be positive under the partial transpose. However, for a density matrix, $W_2$ is always positive semi-definite. This is because you could certainly define a density matrix of the form $$ \left(\begin{array}{cccc} \rho_{00,00} & \rho_{01,00} & 0 & 0 \\ \rho_{00,10} & \rho_{01,01} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$ up to some rescaling. This has the same determinant $W_2$ but is separable, and hence unaffected by the partial transpose. Hence, non-positivity is determined by looking at a violation caused by $W_3$ or $W_4$.

I agree with a query in the comments that it is not immediately obvious that it is relevant for detecting negativity (I will continue to think about it...). If it is a problem, we should be able to come up with a counter-example.

The method is easily extended to any smallest eigenvalue. Simply reduce the matrix by $x$ times the appropriately sized identity matrix, shifting the min eigenvalue from $x$ to 0.