Tangles of superpositions and the convex-roof extension

We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, $\mathrm{SL}(2,\mathbb{C})$ invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the zero polytope is a convex set of density matrices with vanishing tangle whereas the convex characteristic curve readily provides a nontrivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.

Figure 1

(Color online) Characteristic curves $\tau (q,\phi )$ in the $\tau \text{−}q$ plane (solid gray lines) for the superpositions in Eq. (10). The (blue) dashed line represents the minimum $\tilde{\tau }\left(q\right)$ from Eq. (11) and the solid (red) line its function convex hull ${\tau }^{*}\left(q\right)$, i.e., the convex characteristic curve (for details see text). The point with label $\rho$ indicates the average tangle of a particular length-2 decomposition of the density matrix $\rho ={\tilde{p}}_0\mid {\varphi }_0⟩+{\tilde{p}}_1\mid {\varphi }_1⟩$. Moreover we illustrate the actual tangle of this density matrix $\tau \left(\rho \right)$ and its lower bound ${\tau }^{*}\left(p_0\right)$ according to the convex characteristic curve.

Figure 2

(Color online) The 3-tangle of the states $\mid Z(p,\phi )⟩$ in Eq. (17) for various values of $\phi =\gamma \frac{2\pi }{3}$ (from top to bottom: $\gamma =1∕2,1∕3,1∕5,1∕10,0$).

Figure 3

(Color online) Bloch sphere for the two-dimensional space spanned by the GHZ state and the $W$ state. The simplex $S_0$ contains all states with vanishing 3-tangle. The “leaves” between $p_0$ and $p_1$ represent sets of constant 3-tangle. In the simplex $S_1$, the 3-tangle is an affine function (for further explanation, see text).

Figure 4

(Color online) Solid (blue) line: exact solution for the concurrence $C\mathbf{\left(}{\rho }_2\right(p\left)\mathbf{\right)}$ according to Eqs. (29, 30, 31). For comparison: the convex characteristic curve of ${\rho }_2\left(p\right)$ (dashed red line; see also Fig. 5). The exact concurrence $C\mathbf{\left(}\rho \right(p\left)\mathbf{\right)}$ has a zero for ${\tilde{p}}_0=5∕11$.

Figure 5

(Color online) Concurrence for superpositions of states $\mid \mathrm{I}⟩$ and $\mid \mathrm{II}⟩$ (solid gray lines) and convex characteristic curve $C^{*}\left(p\right)$ (dashed red line). The convex characteristic curve coincides with the abscissa [i.e., $C^{*}\left(p\right)\equiv 0$] between the two zeros of Eq. (32), ${\tilde{p}}_1=5∕29$ and ${\tilde{p}}_2=10∕13$.