Bounding Polynomial Entanglement Measures For Mixed States

Peter Love, Haverford College
Samuel Rodriques '13, Class of 2013 Haverford College


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-Eapproximation (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 Sdecomposition 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 d8logd. 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.