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

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

The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on C ⊗ C whose operatorSchmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on C ⊗ C with operatorSchmidt number S for every S ∈ {1, ..., 9}. This corollary...

nt - p h / 03 06 14 4 v 1 2 2 Ju n 20 03 Operator - Schmidt decompositions and the Fourier transform , with applications to the operator - Schmidt numbers of unitaries

The operator-Schmidt decomposition is useful in quantum information theory for quantifying the nonlocality of bipartite unitary operations. We construct a family of unitary operators on C ⊗ C whose operatorSchmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on C ⊗ C with operatorSchmidt number S for every S ∈ {1, ..., 9}. This corollary...

All Unitaries Having Operator Schmidt Rank 2 are Controlled Unitaries

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 di...

Radonification of Cylindrical Semimartingales by a Single Hilbert Schmidt Operator

It is proved that in Hilbert spaces a single Hilbert–Schmidt operator radonifies cylindrical semimartingales to strong semimartingales. This improves a result due to Badrikian and¨Ustünel (also L. Schwartz), who needed composition of three Hilbert–Schmidt operators.