Upper bound for SL-invariant entanglement measures of mixed states

An algorithm is proposed that serves to handle full-rank density matrices when coming from a lower-rank method to compute the convex roof. This is in order to calculate an upper bound for any polynomial SL-invariant multipartite entanglement measure $E$. This study exemplifies how this algorithm works based on a method for calculating convex roofs of rank-2 density matrices. It iteratively considers the decompositions of the density matrix into two states each, exploiting the knowledge for the rank-2 case. The algorithm is therefore quasiexact as far as the rank-2 case is concerned, and it also hints where it should include more states in the decomposition of the density matrix. Focusing on the measure of three-way entanglement of qubits (called three-tangle), I show the results the algorithm gives for two states, one of which is the Greenberger-Horne-Zeilinger–Werner (GHZ-$W$) state, for which the exact convex roof is known. It overestimates the three-tangle in the state, thereby giving insight into the optimal decomposition the GHZ-$W$ state has. As a proof of principle, I have run the algorithm for the three-tangle on the transverse quantum Ising model. I give qualitative and quantitative arguments why the convex roof should be close to the upper bound found here.

Figure 1

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

Figure 2

The graph produced by the algorithm (black curve) is shown together with the convexified curve (blue dashed curve), which gives the correct upper bound to the convex roof. It is hence zero up to the value of $p\approx 0.788675$ and begins to linearly increase to finally reach $\sqrt{{\tau }_3}\left[\right|\phi \rangle ]=1/\sqrt{3}$. At the value of $p=3/4$ it is hence zero, confirming the result of Ref. [14]. The minimal value of $p\approx 0.788675$ is a result of this work.

Figure 3

The graph produced by the algorithm (upper black curve) for the modified basis is shown together with the convex roof of $\sqrt{{\tau }_3}$ (lower orange curve) [13].

Figure 4

The upper bound resulting from the proposed algorithm is shown for the transverse quantum Ising model as a function of $\lambda$. I have furthermore calculated the lower bound of Ref. [14]. It is non-zero only for $0\le \lambda \lesssim 0.9206$ and stays considerably below the upper bound. For $\lambda >0.9206$ it is zero. In particular, it is also the trivial lower bound in the vicinity of the critical point $\lambda =1$.