dorsal/arxiv
View SchemaOperator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries
| Authors | Jon E Tyson |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0306144 |
| URL | https://arxiv.org/abs/quant-ph/0306144 |
| DOI | 10.1088/0305-4470/36/39/309 |
| Journal | J. Phys. A: Math. Gen. 36 (2003) 10101-10114 |
Abstract
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)].
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"abstract": "The operator-Schmidt decomposition is useful in quantum information theory\nfor quantifying the nonlocality of bipartite unitary operations. We construct a\nfamily of unitary operators on C^n tensor C^n whose operator-Schmidt\ndecompositions are computed using the discrete Fourier transform. As a\ncorollary, we produce unitaries on C^3 tensor C^3 with operator-Schmidt number\nS for every S in {1,...,9}. This corollary was unexpected, since it\ncontradicted reasonable conjectures of Nielsen et al [Phys. Rev. A 67 (2003)\n052301] based on intuition from a striking result in the two-qubit case. By the\nresults of Dur, Vidal, and Cirac [Phys. Rev. Lett. 89 (2002) 057901\nquant-ph/0112124], who also considered the two-qubit case, our result implies\nthat there are nine equivalence classes of unitaries on C^3 tensor C^3 which\nare probabilistically interconvertible by (stochastic) local operations and\nclassical communication. As another corollary, a prescription is produced for\nconstructing maximally-entangled operators from biunimodular functions.\nReversing tact, we state a generalized operator-Schmidt decomposition of the\nquantum Fourier transform considered as an operator C^M_1 tensor C^M_2 --\u003e\nC^N_1 tensor C^N_2, with M_1 x M_2 = N_1 x N_2. This decomposition shows (by\nNielsen\u0027s bound) that the communication cost of the QFT remains maximal when a\nnet transfer of qudits is permitted. In an appendix, a canonical procedure is\ngiven for removing basis-dependence for results and proofs depending on the\n\"magic basis\" introduced in [S. Hill and W. Wootters, \"Entanglement of a pair\nof quantum bits,\" Phys Rev. Lett 78 (1997) 5022-5025, quant-ph/9703041 (and\nquant-ph/9709029)].",
"arxiv_id": "quant-ph/0306144",
"authors": [
"Jon E Tyson"
],
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"quant-ph"
],
"doi": "10.1088/0305-4470/36/39/309",
"journal_ref": "J. Phys. A: Math. Gen. 36 (2003) 10101-10114",
"title": "Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries",
"url": "https://arxiv.org/abs/quant-ph/0306144"
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