Abstract
Unitary error bases are fundamental primitives in quantum computing, which are instrumental for quantum error-correcting codes and the design of teleportation and super-dense coding schemes. There are two prominent constructions of such bases: an algebraic construction using projective representations of finite groups and a combinatorial construction using Latin squares and Hadamard matrices. An open problem posed by Schlingemann and Werner relates these two constructions, and asks whether each algebraic construction is equivalent to a combinatorial construction. We answer this question by giving an explicit counterexample in dimension 165 which has been constructed with the help of a computer algebra system.
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Klappenecker, A., Rötteler, M. (2003). Unitary Error Bases: Constructions, Equivalence, and Applications. In: Fossorier, M., Høholdt, T., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2003. Lecture Notes in Computer Science, vol 2643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44828-4_16
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DOI: https://doi.org/10.1007/3-540-44828-4_16
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