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**Published in:** LIPIcs, Volume 22, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)

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.

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)

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@InProceedings{klassen_et_al:LIPIcs.TQC.2013.35, author = {Klassen, Joel and Chen, Jianxin and Zeng, Bei}, title = {{Universal Entanglers for Bosonic and Fermionic Systems}}, booktitle = {8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)}, pages = {35--49}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-55-2}, ISSN = {1868-8969}, year = {2013}, volume = {22}, editor = {Severini, Simone and Brandao, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2013.35}, URN = {urn:nbn:de:0030-drops-43223}, doi = {10.4230/LIPIcs.TQC.2013.35}, annote = {Keywords: Universal Entangler, Bosonic States, Fermionic States} }

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**Published in:** LIPIcs, Volume 22, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)

Symmetry is at the heart of coding theory. Codes with symmetry, especially cyclic codes, play an essential role in both theory and practical applications of classical error-correcting codes. Here we examine symmetry properties for codeword stabilized (CWS) quantum codes, which is the most general framework for constructing quantum error-correcting codes known to date. A CWS code Q can be represented by a self-dual additive code S and a classical code C, i.e., Q=(S,C), however this representation is in general not unique. We show that for any CWS code Q with certain permutation symmetry, one can always find a self-dual additive code S with the same permutation symmetry as Q such that Q=(S,C). As many good CWS codes have been found by starting from a chosen S, this ensures that when trying to find CWS codes with certain permutation symmetry, the choice of S with the same symmetry will suffice. A key step for this result is a new canonical representation for CWS codes, which is given in terms of a unique decomposition as union stabilizer codes. For CWS codes, so far mainly the standard form (G,C) has been considered, where G is a graph state. We analyze the symmetry of the corresponding graph of G, which in general cannot possess the same permutation symmetry as Q. We show that it is indeed the case for the toric code on a square lattice with translational symmetry, even if its encoding graph can be chosen to be translational invariant.

Salman Beigi, Jianxin Chen, Markus Grassl, Zhengfeng Ji, Qiang Wang, and Bei Zeng. Symmetries of Codeword Stabilized Quantum Codes. In 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 22, pp. 192-206, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{beigi_et_al:LIPIcs.TQC.2013.192, author = {Beigi, Salman and Chen, Jianxin and Grassl, Markus and Ji, Zhengfeng and Wang, Qiang and Zeng, Bei}, title = {{Symmetries of Codeword Stabilized Quantum Codes}}, booktitle = {8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)}, pages = {192--206}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-55-2}, ISSN = {1868-8969}, year = {2013}, volume = {22}, editor = {Severini, Simone and Brandao, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2013.192}, URN = {urn:nbn:de:0030-drops-43129}, doi = {10.4230/LIPIcs.TQC.2013.192}, annote = {Keywords: CWS Codes, Union Stabilizer Codes, Permutation Symmetry, Toric Code} }