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 ρ 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 ρ and describe an algorithm to construct the entanglement-minimizing S decomposition for ρ and place an upper bound on E(ρ). 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 d8 log d. 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