Constructions of Unextendible Special Entangled Bases - Wang, Yan-Ling (Frontiers in Physics, 2022)
Unextendible product basis (UPB), a set of incomplete orthonormal product states whose complementary space has no product state, is very useful for constructing bound entangled states. Naturally, instead of considering the set of product states, Bravyi and Smolin considered the set of maximally entangled states. They introduced the concept of unextendible maximally entangled basis (UMEB), a set of incomplete orthonormal maximally entangled states whose complementary space contains no maximally entangled state [Phys. Rev. A \textbf{84}, 042306 (2011)]. An entangled state whose nonzero Schmidt coefficients are all equal to $1/\sqrt{k}$ is called a special entangled state of ``type $k$". In this paper, we introduce a concept named special unextendible entangled basis of ``type $k$" which generalizes both UPB and UMEB. A special unextendible entangled basis of ``type $k$" (SUEB$_k$) is a set of incomplete orthonormal special entangled states of ``type $k$" whose complementary space has no special entangled state of ``type $k$". We present an efficient method to construct sets of SUEB$_k$. The main strategy here is to decompose the whole space into two subspaces such that the rank of each element in one subspace can be easily upper bounded by $k$ while the other one can be generated by two kinds of the special entangled states of ``type $k$". This method is very effective when $k=p^m\geq 3$ where $p$ is a prime number. For these cases, we can obtain sets of SUEB$_k$ with continuous integer cardinality when the local dimensions are large.