Operator-schmidt Decomposition of the Quantum Fourier Transform on C N 1 ⊗ C N 2

Operator-Schmidt decompositions of the quantum Fourier transform on C N1 ⊗ C N2 are computed for all N1,N2 ≥ 2. The decomposition is shown to be completely degenerate when N1 is a factor of N2 and when N1 > N2. The first known special case, N1 = N2 = 2 , 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, PhD Thesis, University of New Mexico (1998), Chapter 6, arXiv:quant-ph/0011036.] More generally, the special case N1 = 2 n1 ≤ 22 = N2 was computed by Nielsen et. al. in their study of strength measures of quantum operations. [M. A. Nielsen et. al, (accepted for publication in Phys. Rev. A); arXiv:quant-ph/0208077.] 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. PACS numbers: 03.67.-a Email: [email protected] current address