  In the realm of quantum information theory, the detection and quantification
of quantum entanglement stand as paramount tasks. The relative entropy of
entanglement (REE) serves as a prominent measure of entanglement, with
extensive applications spanning numerous related fields. The positive partial
transpose (PPT) criterion, while providing an efficient method for the
computation of REE, unfortunately, falls short when dealing with bound
entanglement. In this study, we propose a method termed "pure bosonic
extension" to enhance the practicability of $k$-bosonic extensions, which
approximates the set of separable states from the "outside", through a
hierarchical structure. It enables efficient characterization of the set of
$k$-bosonic extendible states, facilitating the derivation of accurate lower
bounds for REE. Compared to the Semi-Definite Programming (SDP) approach, such
as the symmetric/bosonic extension function in QETLAB, our algorithm supports
much larger dimensions and higher values of extension $k$.
