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Universal Entanglers for Bosonic and Fermionic Systems

Authors Joel Klassen, Jianxin Chen, Bei Zeng



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Joel Klassen
Jianxin Chen
Bei Zeng

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Joel Klassen, Jianxin Chen, and Bei Zeng. Universal Entanglers for Bosonic and Fermionic Systems. In 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 22, pp. 35-49, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)
https://doi.org/10.4230/LIPIcs.TQC.2013.35

Abstract

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 C^{d_1} otimes C^{d_2}, a UE exists when min(d_1,d_2) >= 3 and (d_1,d_2) != (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 C^d otimes 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 C^d otimes C^d if and only if d >= 3 (resp. d >= 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 indistinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.
Keywords
  • Universal Entangler
  • Bosonic States
  • Fermionic States

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