Bounding polynomial entanglement measures for mixed states

We generalize the notion of the best separable approximation (BSA) and best $W$-class approximation (BWA) to arbitrary pure-state entanglement measures, defining the best zero-$E$ approximation (BEA). We show that for any polynomial entanglement measure $E$, any mixed state $\rho$ admits at least one “$S$ decomposition,” i.e., a decomposition in terms of a mixed state on which $E$ is equal to zero, and a single additional pure state with (possibly) nonzero $E$. We show that the BEA is not, in general, the optimal $S$ decomposition from the point of view of bounding the entanglement of $\rho$ and describe an algorithm to construct the entanglement-minimizing $S$ decomposition for $\rho$ and place an upper bound on $E\left(\rho \right)$. When applied to the three-tangle, the cost of the algorithm is linear in the rank $d$ of the density matrix and has an accuracy comparable to a steepest-descent algorithm whose cost scales as $d^{8}logd$. We compare the upper bound to a lower-bound algorithm given by C. Eltschka and J. Siewert [Phys. Rev. Lett. 108, 020502 (2012)] for the three-tangle and find that on random rank-2 three-qubit density matrices, the difference between the upper and lower bounds is $0.14$ on average. We also find that the three-tangle of random full-rank three-qubit density matrices is less than $0.023$ on average.

Figure 1

Upper bounds on the three-tangle of the states in Eq. (15) for 11 values of $p$ (red circles) compared to the analytical value (line). For ten values of $p$ (blue squares) the states in Eq. (15) were conjugated by a random element of $\mathrm{SU}{\left(2\right)}^{\otimes 3}$ sampled from the Gaussian unitary ensemble. For $p=0$ through $p=0.6$, the algorithm calculated the three-tangle to be zero to numerical precision. The inset shows the upper bounds (red dots) on the three-tangle of the density matrices in Eq. (15) for 11 values of $p$ between $0.6$ and $0.7$, compared to the analytical value (line). These are results from 400 repetitions of the upper-bound algorithm.

Figure 2

Cumulative distribution function of the upper bound on the three-tangle of 10 000 randomly sampled three-qubit density matrices of rank 2 (lower solid line), 4 (dotted line), 6 (dashed line), and 8 (upper solid line). The inset shows the cumulative distribution function of the difference between the upper and lower bounds on the three-tangle of 30 000 uniformly sampled three-qubit rank-2 density matrices.