Operator-Schmidt decomposition of the quantum Fourier transform on N1 N2
Jon E. Tyson · Journal of Physics A Mathematical and General · 2003
Operator-Schmidt decompositions of the quantum Fourier transform on ℂ N 1 ⊗ ℂ N 2 are computed for all N 1 , N 2 ≥ 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 n 1 ⩽ 2 n 2 = 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.