Detecting entanglement by pure bosonic extension

Detecting and quantifying quantum entanglement is a central task in quantum information theory. Relative entropy of entanglement (REE) is one of the most famous quantities for measuring entanglement and has various applications in many other fields. One well-studied and efficient approach for calculating the lower bound of REE is the positive partial transpose (PPT) criterion. But it fails in the bound entangled area. In this work, we use a method called pure bosonic extension to significantly improve the feasibility of k-symmetric/bosonic extensions which characterize the separable set from outside by a hierarchy structure. Based on this method, we can efficiently approximate the boundaries of k-bosonic extendible sets and obtain the desired lower bound of REE. Compared to the Semi-Definite Programming method, for example, the symmetric extension function in QETLAB, our algorithm can support much larger single particle dimensions and much larger k.