Entanglement witness

Remarks

Example: let $H_A = H_B = \C^2$ and let $W = F$ be the swap operator, $F(u \otimes v) = v \otimes u$. For a product state, $\tr(F \, \sigma_A \otimes \sigma_B) = \tr(\sigma_A \sigma_B) \ge 0$, and clause 1 follows by linearity. For the singlet $\psi = (|01\rangle - |10\rangle)/\sqrt{2}$, one has $F\psi = -\psi$, so $\tr(F |\psi\rangle\langle\psi|) = -1$ and clause 2 holds.

Non-example: the identity $I$ satisfies clause 1 but fails clause 2, since $\tr(I\rho) = 1$ for every state. The operator $-F$ fails clause 1, since $\tr(-F \, |00\rangle\langle 00|) = -1$.

Since the separable states form a compact convex set, the separating hyperplane theorem gives: for every entangled state $\rho$ there is a witness $W$ that detects $\rho$.