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arXiv:quant-ph/0210100 · Journal of Physics A Mathematical and General

Operator-Schmidt decomposition of the quantum Fourier transform on N1 N2

Jon Tyson

Published Jun 5, 2003• 17 citations
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Abstract

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.

DOI
10.1088/0305-4470/36/24/317

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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.
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^n tensor C^n whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on C^3 tensor C^3 with operator-Schmidt number S for every S in {1,...,9}. This corollary was unexpected, since it contradicted 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 the results of Dur, Vidal, and Cirac [Phys. Rev. Lett. 89 (2002) 057901 quant-ph/0112124], who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on C^3 tensor C^3 which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled operators from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum 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 (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the "magic basis" introduced in [S. Hill and W. Wootters, "Entanglement of a pair of quantum bits," Phys Rev. Lett 78 (1997) 5022-5025, quant-ph/9703041 (and quant-ph/9709029)].
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 n ⊗ n whose operator-Schmidt decompositions are computed using the discrete Fourier transform. As a corollary, we produce unitaries on 3 ⊗ 3 with operator-Schmidt number S for every S ∊ {1, ..., 9}. This corollary was unexpected, since it contradicted reasonable conjectures of Nielsen et al (2003 Phys. Rev. A 67 052301) based on intuition from a striking result in the two-qubit case. By the results of Dür et al (2002 Phys. Rev. Lett. 89 057901), who also considered the two-qubit case, our result implies that there are nine equivalence classes of unitaries on 3 ⊗ 3 which are probabilistically interconvertible by (stochastic) local operations and classical communication. As another corollary, a prescription is produced for constructing maximally-entangled unitaries from biunimodular functions. Reversing tact, we state a generalized operator-Schmidt decomposition of the quantum Fourier transform considered as an operator M 1 ⊗ M 2 → N 1 ⊗ N 2 , with M 1 M 2 = N 1 N 2 . This decomposition shows (by Nielsen's bound) that the communication cost of the QFT remains maximal when a net transfer of qudits is permitted. In an appendix, a canonical procedure is given for removing basis-dependence for results and proofs depending on the 'magic basis' introduced in S Hill and W Wootters (1997 Entanglement of a pair of quantum bits Phys Rev. Lett. 78 5022–5).
It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we derive several generally valid purity-type formulas for one-party reductions of rank-1 operators, and we point out relevant relations between the Schmidt decomposition and the Bloch representation of bipartite pure states.
The Quantum Fourier transform (QFT) is a key ingredient in most quantum algorithms. We have compared various spin-based quantum computing schemes to implement the QFT from the point of view of their actual time-costs and the accuracy of the implementation. We focus here on an interesting decomposition of the QFT as a product of the non-selective Hadamard transformation followed by multiqubit gates corresponding to square- and higher-roots of controlled-NOT gates. This decomposition requires only O(n) operations and is thus linear in the number of qubits n. The schemes were implemented on a two-qubit NMR quantum information processor and the resultant density matrices reconstructed using standard quantum state tomography techniques. Their experimental fidelities have been measured and compared.
The Quantum Fourier transform (QFT) is a key ingredient in most quantum algorithms. We have compared various spin-based quantum computing schemes to implement the QFT from the point of view of their actual time-costs and the accuracy of the implementation. We focus here on an interesting decomposition of the QFT as a product of the non-selective Hadamard transformation followed by multiqubit gates corresponding to square- and higher-roots of controlled-NOT gates. This decomposition requires only O(n) operations and is thus linear in the number of qubits n. The schemes were implemented on a two-qubit NMR quantum information processor and the resultant density matrices reconstructed using standard quantum state tomography techniques. Their experimental fidelities have been measured and compared.