Unextendible Product Basis for Fermionic Systems

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.Comment: 17 pages, no figure. Comments are welcom

The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)

A long-standing open question asks for the minimum number of vectors needed to form an unextendible product basis in a given bipartite or multipartite Hilbert space. A partial solution was found by Alon and Lovasz in 2001, but since then only a few other cases have been solved. We solve all remaining bipartite cases, as well as a large family of multipartite cases.Comment: 17 pages, 4 figure

Universal Entanglers for Bosonic and Fermionic Systems

A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles $1,2$ with a Hilbert space $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$, a UE exists when $\min{(d_1,d_2)}\geq 3$ and $(d_1,d_2)\neq (3,3)$. It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic systems whose wave functions sit in the symmetric/antisymmetric subspace of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. The development of a theory of UEs for these types of systems needs considerably different approaches from that used for UEs of distinguishable systems. This is because the general entanglement of identical particle systems cannot be discussed in the usual way due to the effect of (anti)-symmetrization which introduces "pseudo entanglement" that is inaccessible in practice. We show that, unlike the distinguishable particle case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are symmetric (resp. antisymmetric) subspaces of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ if and only if $d\geq 3$ (resp. $d\geq 8$). To prove this we employ algebraic geometry to reason about the different algebraic structures of the bosonic/fermionic systems. Additionally, due to the relatively simple coherent state form of unentangled bosonic states, we are able to give the explicit constructions of two bosonic UEs. Our investigation provides insight into the entanglement properties of systems of indisitinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.Comment: 15 pages, comments welcome, TQC2013 Accepted Tal