1,675 research outputs found

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

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    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

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    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 1,21,2 with a Hilbert space Cd1⊗Cd2\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}, a UE exists when min⁡(d1,d2)≥3\min{(d_1,d_2)}\geq 3 and (d1,d2)≠(3,3)(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 Cd⊗Cd\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 Cd⊗Cd\mathbb{C}^{d}\otimes\mathbb{C}^{d} if and only if d≥3d\geq 3 (resp. d≥8d\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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