Archipelagos of Total Bound and Free Entanglement

First, we considerably simplify an initially quite complicated formula – involving dilogarithms. It yields the total bound entanglement probability ($\approx 0.0865542$) for a qubit-ququart ($2\times 4$) three-parameter model, recently analyzed for its separability properties by Li and Qiao. An “archipelago” of disjoint bound-entangled regions appears in the space of parameters, somewhat similarly to those recently found in our preprint, “Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities”. There, two-qutrit and two-ququart Hiesmayr-L{"o}ffler “magic simplices” and generalized Horodecki states had been examined. However, contrastingly, in the present study, the entirety of bound entanglement–given by the formula obtained–is clearly captured in the archipelago found. Further, we “upgrade” the qubit-ququart model to a two-ququart one, for which we again find a bound-entangled archipelago, with its total probability simply being now $\frac{1}{729} \left(473-512 \log \left(\frac{27}{16}\right) \left(1+\log \left(\frac{27}{16}\right)\right)\right) \approx 0.0890496$. Then, “downgrading” the qubit-ququart model to a two-qubit one, we find an archipelago of total non-bound/free entanglement probability $\frac{1}{2}$.