@article {1445, title = {Unextendible Product Basis for Fermionic Systems}, journal = {Journal of Mathematical Physics}, volume = {55}, year = {2014}, month = {2014/01/01}, pages = {082207}, abstract = { We discuss the concept of unextendible product basis (UPB) and generalized UPB for fermionic systems, using Slater determinants as an analogue of product states, in the antisymmetric subspace $\wedge^ N \bC^M$. We construct an explicit example of generalized fermionic unextendible product basis (FUPB) of minimum cardinality $N(M-N)+1$ for any $N\ge2,M\ge4$. We also show that any bipartite antisymmetric space $\wedge^ 2 \bC^M$ of codimension two is spanned by Slater determinants, and the spaces of higher codimension may not be spanned by Slater determinants. Furthermore, we construct an example of complex FUPB of $N=2,M=4$ with minimum cardinality $5$. In contrast, we show that a real FUPB does not exist for $N=2,M=4$ . Finally we provide a systematic construction for FUPBs of higher dimensions using FUPBs and UPBs of lower dimensions. }, doi = {10.1063/1.4893358}, url = {http://arxiv.org/abs/1312.4218v1}, author = {Jianxin Chen and Lin Chen and Bei Zeng} }