Detecting entanglement by pure bosonic extension

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>k</a:mi></a:math>-bosonic extensions, which approximates the set of separable states from the “outside”, through a hierarchical structure. It enables efficient characterization of the set of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>k</b:mi></b:math>-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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>k</c:mi></c:math>.