Three-tangle of the nine classes of four-qubit states

I calculate the mixed three-tangle ${\tau }_3\left[\rho \right]$ for the reduced density matrices of the four-qubit representant states found in Phys. Rev. A 65, 052112 (2002). In most of the cases, the convex roof is obtained, except for one class, where I provide with an upper bound, which is assumed to be very close to the convex roof. I compare with results published in Phys. Rev. Lett. 113, 110501 (2014). Since the method applied there usually results in higher values for the upper bound, in certain cases it can be understood that the convex roof is obtained exactly, namely, when the zero-polytope where ${\tau }_3$ vanishes shrinks to a single point.

Figure 1

The Bloch sphere with radius $|\vec{r}|=1$ and the standard parametrization of its surface in the angles $\vartheta$ and $\phi$ of the vector $\vec{r}$. This vector corresponds to the pure state $|\mathrm{\Psi }(p_1;\phi )\rangle =cos\frac{\vartheta \left(p_1\right)}{2}|{\psi }_1\rangle +sin\frac{\vartheta \left(p_1\right)}{2}{e}^{i\phi }|{\psi }_2\rangle =\sqrt{p_1}|{\psi }_1\rangle +\sqrt{1-p_1}{e}^{i\phi }|{\psi }_2\rangle$. The $r_{\mathfrak{z}}$ component is related to the probability $p_1$ via $r_{\mathfrak{z}}=2p_1-1$.

Figure 2

The minimal characteristic curves for the angles $\phi =arg{z}_0+j\pi$ on the Bloch sphere are shown for $j=0,1$ (black line) together with the convexified curve ${\tau }_3^{\mathrm{conv}}$ (blue dashed line). The optimal decompositions are $|{\psi }_2\rangle$ and the two corresponding $|{\psi }_{z_0}\rangle$ for a finite value of $\sqrt{{\tau }_3}$, and $|{\psi }_1\rangle$ and the two $|{\mathrm{\Psi }}_{z_0}\rangle$ for vanishing three-tangle (see explanations in the text).

Figure 3

A typical example for a minimal characteristic curve for ${\tau }_3$ is shown (black line) together with the convexified one ${\tau }_3^{\mathrm{conv}}$ (blue dashed line). Here the curve is strictly convex where the three-tangle is finite. This leads to an optimal decomposition at $p_1$ which is $|{\mathrm{\Psi }}_{\pm z\left(p_1\right)}\rangle$ and gives rise to smaller values for the convex roof ${\widehat{\tau }}_3$.

Figure 4

Class 2: The convex roofs are shown for ${\widehat{\tau }}_3$ (central orange curve) and ${\widehat{\sqrt{{\tau }_3}}}^2$ (lower purple curve) for real values of $a=0.65, b=0.35$, and $c$. It is seen that ${\widehat{\sqrt{{\tau }_3}}}^2<{\widehat{\tau }}_3$. For comparison we plot also the curve obtained by Ref. [28] from the algorithm suggested in Ref. [26] (upper black curve). They coincide at the pikes of the curves, where the convex roof is given by the eigen-decomposition.

Figure 5

Class 3: The convex roofs $\widehat{{\tau }_3}$ (central orange curve) and ${\widehat{\sqrt{{\tau }_3}}}^2$ (lower purple curve) are shown together with the estimate of Ref. [28] (upper black curve) for $a=2.0$. It can be seen that they both coincide at $a=b=2.0$, where the minimal characteristic curve of $\widehat{\sqrt{{\tau }_3}}$ is a straight line. At this point, it also must coincide with the estimate in Ref. [28], which otherwise highly overestimates the convex roof. The optimal decompositions are made out of $|{\mathrm{\Psi }}_{\pm z_0}\rangle$ and $|{\psi }_2\rangle$ for a finite $\widehat{\sqrt{{\tau }_3}}$, and $|{\mathrm{\Psi }}_{\pm z\left(p_1\right)}\rangle$ for finite ${\widehat{\tau }}_3$.

Figure 6

Some characteristic curves are shown for real values $a=0.4, b=1.9$ and angles $\phi =\pi$ (the minimum; lower purple curve), $\phi =\pm \pi /2$ (central orange curve), and $\phi =0$ (the maximum; upper blue curve). For comparison the straight line (thin black curve) marks the convex combination of the two eigenstates. There is a single zero of fourth order at $z=-\left|\right|{\psi }_2\rangle |/\left|\right|{\psi }_1\rangle |$ of the equation ${\tau }_3\left(\right|{\psi }_1\rangle /\left|\right|{\psi }_1\rangle |+z|{\psi }_2\rangle /\left|\right|{\psi }_2\rangle \left|\right)=0$.

Figure 7

The minimal (lower purple curve) and maximal (upper blue curve) characteristic curves are shown here for real $a=0.4$ and $b=1.9$ and the same angles as in Fig. 6. The curves at $\phi =\pm \pi /2$ coincide and even give the convex roof result. Here this is a straight line connecting both eigenstates, and consequently the algorithm of Ref. [26] also gives this result.

Figure 8

The values for $\widehat{{\tau }_3}={\widehat{\sqrt{{\tau }_3}}}^2$ in class 4 for real values of $a$ and $b$.

Figure 9

For real value $a=0.3$ among some characteristic curves are shown in particular the minimal characteristic curve at $\phi =\pi /2$ (lower blue line) and maximal characteristic curve at $\phi =3\pi /2$ (higher red curve) and the two intermediate coinciding curves at $\phi =0,\pi$ (central orange curve).

Figure 10

For $a=0.3$ the sum of two by an angle of $\pi$ differing characteristic curves is shown for $\phi =0,\pi$ (higher orange curve) and for $\phi =\pi /2,3\pi /2$ (lower purple curve), combining to possible decompositions of $\rho$. The convexification of this latter curve is also shown (purple dashed line). A decomposition into three states (thin red curve) is shown to be situated between the lower purple and higher orange curve.

Figure 11

In addition to Fig. 10, convex combinations of the state $|{\mathrm{\Psi }}_{z_0}\rangle$ with a second state at $\phi =\pi /2$ to give a decomposition of $\rho \left(p_1\right)$ are shown in the convex lower black curve. Every combination from the zero-simplex with another state at $\phi =\pi /2$ lies above this curve. This is the lowest upper bound. I conjecture that it equals the convex roof.

Figure 12

Class 5 for qubits 1 or 3 traced out: Upper bounds for ${\widehat{\sqrt{{\tau }_3}}}^2$ (which is the conjectured convex roof; lower blue curve) and ${\widehat{\tau }}_3$ (central orange curve) are shown. They are compared with the result in Ref. [28] (upper black curve) making use of the algorithm related to Ref. [26]. It is seen that it overestimates the value for $\widehat{\sqrt{{\tau }_3}}$. It astonishingly coincides for small values of $a$ with the upper bound for ${\widehat{\tau }}_3$ but then deviates from it. This coincidence is due to not having calculated the upper bound for the convexification of the corresponding minimal curve that I have found. It is nicely seen that the upper bounds satisfy the inequality ${\widehat{\sqrt{{\tau }_3}}}^2<{\widehat{\tau }}_3$ for the convex roofs.

Figure 13

Class 6: Both convex roofs ${\widehat{\sqrt{{\tau }_3}}}^2$ (blue lower curve) and ${\widehat{\tau }}_3$ (orange curve) are shown. It is seen that ${\widehat{\sqrt{{\tau }_3}}}^2\le {\widehat{\tau }}_3$. Both values vanish if $\left|a\right|\ge 2^{2/3}$. It is strange that the upper bound from Ref. [28] coincides with ${\widehat{\sqrt{{\tau }_3}}}^2$, where instead something larger would be expected. This appears to be accidental and could be interrelated with only two apparent solutions to the zero-simplex, when the $z$ appears in front of a state with zero three-tangle.