The exact convex roof for GHZ-W mixtures for three qubits and beyond

TL;DR

It addresses the convex roof of the SL-invariant entanglement monotone given by the square root of the threetangle, $\\sqrt{\\tau_3}$, for all states in the three-qubit Bloch ball. It introduces zero-state locking and derives an inequality to determine whether decompositions of type $(n_z,n_e)$ with $n_e>1$ can be optimal, showing that optimal decompositions are constrained to at most four pure states forming 3D simplices. The GHZ-W mixture is analyzed, revealing a characteristic surface pattern on the Bloch sphere: a zero-polytope, four face polytopes, and a network of $(2,1)$ decompositions along grand circles with $(3,1)$ states at the tips; the remaining regions use $(1,1)$ decompositions. $SL$-invariance is used to lift the solution to all $SL$-equivalent states, and the paper outlines a general framework for arbitrary SL-invariant tangles in rank-two mixtures and discusses potential generalizations to higher ranks and practical quantum technologies.

Abstract

I present an exact solution for the convex roof of the square root of the threetangle for all states within the Bloch sphere. The working horse that optimal decompositions contain as many states from the zero-polytope as possible which can be called zero-state locking is proved and an inequality is derived which decides about the optimality of the decompositions under consideration here. The footprint of the measure of entanglement consists in a characteristic pattern for the fixed pure states on the surface which form the optimal solution. The solution is subject to transformation properties due to the SL-invariance of the entanglement measure.

Figures (7)

• Figure 1: Optimal decompositions are obtained for the rank-two mixtures of GHZ and W state and are depicted here graphically. It consists of the zero polytope (blue shaded), 4 polytopes defined by three of the zero states together with a state $| N \rangle$ one of which is the GHZ state itself (grey shaded); all these polytopes are three dimensional. On top of this kernel structure of five polytopes are three lines along grand circles along $\phi$ direction and three lines which are on circles with a distance $0.0711148$ to the center of the Bloch sphere. Along these lines move pure states which form a two-dimensional optimal decomposition (red triangles). Blue points mark the position of the respective normal vectors on the Bloch sphere surface; blue lines the corresponding pure states in $(2,1)$ decompositions. All remaining parts of the Bloch sphere are covered by $(1,1)$ decompositions.
• Figure 2: Left: The zero-polytope is shown (blue) in the Bloch sphere projection in $(x,z)$ plane, together with the sphere of radius $r_0$ given by the two complex conjugated yero states. The entangled pure states marking optimal decompositions are marked by green dots. The plane inclined by an angle $\delta_1$ is shown in purple color, cutting both spheres in circles of radii $d_0$ and $d=d_0+d_1$. The inclination angle is limited by the three-dimensional polytopes by the interval $[\delta_-,\delta_++\pi/2]$. $\delta_-$ is negative. Right: The purple plane inclined by the angle $\delta$ in the Bloch sphere is shown with part of the circles of radii $d_0$ and $d=d_0+d_1$. The density matrix $\rho$ decomposed of the two complex conjugated states located in $(x,\pm y)$ must be better than the $(2,1)$ decomposition in consideration. For any $(2,2)$ decomposition being optimal for a state $\rho'$ located at distance $dr$ from the point $z_0$, there need to be states $\rho$ that have to be optimally decomposed by two corresponding states in $(x,\pm y)$.
• Figure 3: Difference $\Delta\tau:=\tau_{(0,2)}-\tau_{(2,1)}$ of the (0,2) and $(2,1)$ decompositions is shown for $\tau:=\sqrt{\tau_3}$ is shown depending on $\delta_1$ for $p$ close to $1$.
• Figure 4: Left: States within the Bloch sphere. For a given line $\overline{x_1x_2}$ all triangles $z_0x_1z_1$, $z_0x_2z_2$, and $z_0x_2'z_2'$, are similar. Ratios of their length are determined by the theorem of intersecting lines. Right: The optimal decompositions in the Bloch sphere spanned by $GHZ$ and $W$ state.
• Figure 5: Convexification procedure of $| Z3 \rangle$ mixed with the corresponding orthogonal state. The curves are made convex with a line touching at $p_c=0.0964142$, as is highlighted in the inset, were we plot the difference in $\sqrt{\tau_3}$ of the $(1,1)$ decompositions with $| Z_3 \rangle$ and the convex line. The $(2,1)$ decompositions with $| Z_3 \rangle$ and $| W \rangle$ are shown in blue above both curves.