Bound entangled states with positive partial transposition based on UPB in three-qudit system

Abstract In this work, we concentrate on identifying bound entangled states with positive partial transposition in tripartite system. Since directly extending PPT-BE states from bipartite to tripartite systems is nontrivial, we adopt unextendible product bases (UPBs) as a tool to generate tripartite PPT-BE states. First, a set of UPB with cardinalities ${{d}^{3}}-4d+3$ is presented to characterize the extremal states of PPT set in the ${{C}^{d}}\otimes {{C}^{d}}\otimes {{C}^{d}}$ system. Using this basis, the missing states are introduced and employed to identify a $2\left( d-1 \right)$-parameter family of PPT entangled edge states. Additionally, a generation circuit is provided which transforms the states in Z-basis to the missing states in three-qudit composite system. This circuit intuitively reflects the structure of the missing states and prepares for the generation of PPT entangled states. Our work demonstrates a different construction method of PPT entangled edge states in higher-dimensional multipartite system, providing a general insight into the relationship between nonlocality and entanglement in such system.