Universal Entanglers for Bosonic and Fermionic Systems

TitleUniversal Entanglers for Bosonic and Fermionic Systems
Publication TypeJournal Article
Year of Publication2013
AuthorsKlassen, J, Chen, J, Zeng, B
Date Published2013/05/31

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.