Optimal entanglement witnesses for qubits and qutrits

We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separable state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular two-qubit and two-qutrit states—and their generalizations to arbitrary dimensions—where we calculate the optimal entanglement witnesses explicitly.

Figure 1

Geometric illustration of a plane in Euclidean space and the different values of the scalar product for states above $\left({\vec{b}}_u\right)$, within $\left({\vec{b}}_p\right)$, and below the plane $\left({\vec{b}}_d\right)$.

Figure 2

Illustration of an optimal entanglement witness

Figure 3

Illustration of Eqs. (30, 31): The scalar product $⟨{\rho }_l-{\rho }_1,{\rho }_1-{\rho }_2⟩$ is negative because the projection ${({\rho }_l-{\rho }_1)}_{\Vert }$ onto ${\rho }_1-{\rho }_2$ points in the opposite direction to ${\rho }_1-{\rho }_2$. On the other side, $⟨{\rho }_r-{\rho }_1,{\rho }_1-{\rho }_2⟩$ is positive for states ${\rho }_r$, because then the projection ${({\rho }_r-{\rho }_1)}_{\Vert }$ points in the same direction as ${\rho }_1-{\rho }_2$.

Figure 4

Illustration of the Bertlmann-Narnhofer-Thirring theorem

Figure 5

Illustration why $\tilde{C}$ cannot be an entanglement witness if $\tilde{\rho }$ is not the nearest separable state. The hatched area is the one where the condition $⟨\rho ,\tilde{C}⟩⩾0\forall \rho ∊S$ is violated.