Geometric entanglement witnesses and bound entanglement

We study entanglement witnesses that can be constructed with regard to the geometrical structure of the Hilbert-Schmidt space, i.e., we present how to use these witnesses in the context of quantifying entanglement and the detection of bound entangled states. We give examples for a particular three-parameter family of states that are part of the magic simplex of two-qutrit states.

Figure 1

Sketch of the presented method to detect bound entanglement with the geometric entanglement witness $C_{\lambda }$. The dashed line indicates the detected bound entangled states ${\rho }_{\lambda }$ $({\lambda }_{\min }<\lambda ⩽1)$, states ${\rho }_{\lambda }$ with $0<\lambda ⩽{\lambda }_{\min }$ can be separable or bound entangled (straight line).

Figure 2

(Color online) Illustration of states ${\rho }_{\alpha ,\beta }$ (8) $(\gamma =0)$ and their partial transposition. The regions I and II label the regions where the states are NPT entangled, they are PPT and separable in the overlap with the region of PPT points. The PPT points become semipositive under partial transposition.

Figure 3

(Color online) The states ${\rho }_{\alpha ,\beta ,\gamma }$ are depicted with slices of the three-dimensional parameter space for fixed values of $\gamma$ (values of $\gamma$ from $\gamma =0$ down are shown from right to left in the figure). The curved borders of the PPT regions for $\gamma \ne 0$ are hyperbolae that result from intersecting a cone of PPT points with the planes of constant $\gamma$. The slices for positive and negative $\gamma$ overlap. All parameter axes are chosen non-orthogonal such that they become orthogonal to the boundary of the positivity region in order to reproduce the symmetry of the magic simplex.