Hilbert-Schmidt distance and entanglement witnessing

Gilbert proposed an algorithm for bounding the distance between a given point and a convex set. We apply the Gilbert's algorithm to get an upper bound on the Hilbert-Schmidt distance between a given state and the set of separable states. While Hilbert-Schmidt distance does not form a proper entanglement measure, it can nevertheless be useful for witnessing entanglement. We provide a few methods based on the Gilbert's algorithm that can reliably qualify a given state as strongly entangled or practically separable, while being computationally efficient. The method also outputs successively improved approximations to the closest separable state for the given state. We demonstrate the efficacy of the method with examples.

Figure 1

Visualization of the Gilbert algorithm. A random separable state ${\rho }_2$ is generated that satisfies the preselection criterion. The next step is to find ${\rho }_1^{\prime }=p{\rho }_1+(1-p){\rho }_2$, $0\le p\le 1$, such that the new Hilbert-Schmidt distance $D_{01}^{\prime }=D^2({\rho }_0,{\rho }_1^{\prime })$ is less than the previous, $D_{01}=D^2({\rho }_0,{\rho }_1)$. If such a state is found, the state ${\rho }_1^{\prime }$ is updated as the new ${\rho }_1$.

Figure 2

Convergence of $D^2({\rho }_0,{\rho }_1)$ (y axis) to the analytical limit (dashed lines) as a function of number of corrections $c_s$ (x axis) for $d=2,3,\cdots ,9$ (bottom to top).

Figure 3

Relation between $a$ and $b$ found in maximizing $R(x,y)$ in 21 runs of the algorithm for the maximally entangled state of two ququarts ($d=4$) and HALT condition $c_s=1000$. In general, underestimation of the HSD is related to slow initial convergence (high values of $b$), while overestimation is accompanied by fast convergence.

Figure 4

Plot of the minimum distance $D^2({\rho }_{\mathrm{Werner}},{\rho }_1)$ (solid red line) for $d=2$ and expectation of CHSH operator (with 1 being the maximal local realistic values; dashed blue line). The CHSH inequality is violated after the minimum distance becomes nonzero.

Figure 5

Plot of minimum $D^2({\rho }_{\mathrm{Werner}},{\rho }_1)$ for the $d=3$ Werner state (solid red line) and expectation of CGLMP operator (with 2 being the maximal local realistic values; dashed blue line) with the visibility parameter $p$.

Figure 6

Plot of minimum $D^2({\rho }_{\mathrm{Werner}},{\rho }_1)$ for the $d=4$ Werner state (solid red line) and expectation of CGLMP operator (with 2 being the maximal local realistic values; dashed blue line) with the visibility parameter $p$.

Figure 7

Dependence between the number of corrections to ${\rho }_1$ and $D^2({\rho }_0,{\rho }_1)$ for the two-qutrit UPB BE state [19].