2,101 research outputs found

### 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

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