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A universal entangler (UE) is a unitary operation which maps all pure productstates to entangled states. It is known that for a bipartite system ofparticles $1,2$ 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 alsoknown that whenever a UE exists, almost all unitaries are UEs; however toverify whether a given unitary is a UE is very difficult since solving aquadratic system of equations is NP-hard in general. This work examines theexistence and construction of UEs of bipartite bosonic/fermionic systems whosewave functions sit in the symmetric/antisymmetric subspace of$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. The development of a theory of UEs forthese types of systems needs considerably different approaches from that usedfor UEs of distinguishable systems. This is because the general entanglement ofidentical particle systems cannot be discussed in the usual way due to theeffect of (anti)-symmetrization which introduces "pseudo entanglement" that isinaccessible in practice. We show that, unlike the distinguishable particlecase, UEs exist for bosonic/fermionic systems with Hilbert spaces which aresymmetric (resp. antisymmetric) subspaces of$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ if and only if $d\geq 3$ (resp. $d\geq8$). To prove this we employ algebraic geometry to reason about the differentalgebraic structures of the bosonic/fermionic systems. Additionally, due to therelatively simple coherent state form of unentangled bosonic states, we areable to give the explicit constructions of two bosonic UEs. Our investigationprovides insight into the entanglement properties of systems ofindisitinguishable particles, and in particular underscores the differencebetween the entanglement structures of bosonic, fermionic and distinguishableparticle systems.

Universal Entanglers for Bosonic and Fermionic Systems