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The operator-Schmidt decomposition is useful in quantum information theoryfor quantifying the nonlocality of bipartite unitary operations. We construct afamily of unitary operators on C^n tensor C^n whose operator-Schmidtdecompositions are computed using the discrete Fourier transform. As acorollary, we produce unitaries on C^3 tensor C^3 with operator-Schmidt numberS for every S in {1,...,9}. This corollary was unexpected, since itcontradicted reasonable conjectures of Nielsen et al [Phys. Rev. A 67 (2003)052301] based on intuition from a striking result in the two-qubit case. By theresults of Dur, Vidal, and Cirac [Phys. Rev. Lett. 89 (2002) 057901quant-ph/0112124], who also considered the two-qubit case, our result impliesthat there are nine equivalence classes of unitaries on C^3 tensor C^3 whichare probabilistically interconvertible by (stochastic) local operations andclassical communication. As another corollary, a prescription is produced forconstructing maximally-entangled operators from biunimodular functions.Reversing tact, we state a generalized operator-Schmidt decomposition of thequantum Fourier transform considered as an operator C^M_1 tensor C^M_2 -->C^N_1 tensor C^N_2, with M_1 x M_2 = N_1 x N_2. This decomposition shows (byNielsen's bound) that the communication cost of the QFT remains maximal when anet transfer of qudits is permitted. In an appendix, a canonical procedure isgiven for removing basis-dependence for results and proofs depending on the"magic basis" introduced in [S. Hill and W. Wootters, "Entanglement of a pairof quantum bits," Phys Rev. Lett 78 (1997) 5022-5025, quant-ph/9703041 (andquant-ph/9709029)].

Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries