Unextendible product basis for fermionic systems
Jianxin Chen, Lin Chen, Bei Zeng · Journal of Mathematical Physics · 2014
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 anti-symmetric subspace \documentclass[12pt]{minimal}\begin{document}$\wedge ^ N {\mathbb {C}}^M$\end{document}∧NCM. We construct an explicit example of generalized fermionic unextendible product basis (FUPB) with minimum cardinality N(M − N) + 1 for any N ⩾ 2, M ⩾ 4. We also show that any bipartite anti-symmetric space \documentclass[12pt]{minimal}\begin{document}$\wedge ^ 2 {\mathbb {C}}^M$\end{document}∧2CM 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 by using FUPBs and UPBs of lower dimensions.