Operator-Schmidt decomposition of the quantum Fourier transform on CN1 ⊗ CN2

Operator-Schmidt decompositions of the quantum Fourier transform on C-N1 circle times C-N2 are computed for all N-1, N-2 greater than or equal to, 2. The decomposition is shown to be completely degenerate when N-1 is a factor of N-2 and when N-1 > N-2. The first known special case, N-1 = N-2 = 2(n), was computed by Nielsen in his study of the communication cost of computing the quantum Fourier transform of a collection of qubits equally distributed between two parties (M A Nielsen 1998 PhD Thesis University of New Mexico ch 6 Preprint quant-ph/0011036). More generally, the special case N-1 = 2(n1) less than or equal to 2(n2) = N-2 was computed by Nielsen et al in their study of strength measures of quantum operations (M A Nielsen et al 2002 Preprint quant-ph/0208077 (2003 Phys. Rev. A at press)). Given the Schmidt decompositions presented here, it follows that in all cases the bipartite communication cost of exact computation of the quantum Fourier transform is maximal.