Quasirandom estimations of two-qubit operator-monotone-based separability probabilities
Abstract
We conduct a pair of quasirandom estimations of the separability probabilities with respect to 10 measures on the 15-dimensional convex set of two-qubit states, using its Euler-angle parametrization. The measures include the (nonmonotone) Hilbert–Schmidt one, plus nine others based on operator monotone functions. Our results are supportive of previous assertions that the Hilbert–Schmidt and Bures (minimal monotone) separability probabilities are 833≈0.242424833≈0.242424 and 25341≈0.073313825341≈0.0733138, respectively, as well as suggestive of the Wigner–Yanase counterpart being 120120. However, one result appears inconsistent (much too small) with an earlier claim of ours that the separability probability associated with the operator monotone (geometric-mean) function √x√x is 1−25627π2≈0.03932511−25627π2≈0.0393251. But a seeming explanation for this disparity is that the volume of states for the √x√x-based measure is infinite. So, the validity of the earlier conjecture — as well as an alternative one, 19(593−60π2)≈0.091526219(593−60π2)≈0.0915262, we now introduce — cannot be examined through the numerical approach adopted, at least perhaps not without some truncation procedure for extreme values.
