dorsal/arxiv
View SchemaTime Reversal and n-qubit Canonical Decompositions
| Authors | Stephen S. Bullock, Gavin K. Brennen, Dianne P. O'Leary |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0402051 |
| URL | https://arxiv.org/abs/quant-ph/0402051 |
| DOI | 10.1063/1.1900293 |
| Journal | Journal of Mathematical Physics, volume 46, 062104 (2005) |
Abstract
For n an even number of qubits and v a unitary evolution, a matrix decomposition v=k1 a k2 of the unitary group is explicitly computable and allows for study of the dynamics of the concurrence entanglement monotone. The side factors k1 and k2 of this Concurrence Canonical Decomposition (CCD) are concurrence symmetries, so the dynamics reduce to consideration of the a factor. In this work, we provide an explicit numerical algorithm computing v=k1 a k2 for n odd. Further, in the odd case we lift the monotone to a two-argument function, allowing for a theory of concurrence dynamics in odd qubits. The generalization may also be studied using the CCD, leading again to maximal concurrence capacity for most unitaries. The key technique is to consider the spin-flip as a time reversal symmetry operator in Wigner's axiomatization; the original CCD derivation may be restated entirely in terms of this time reversal. En route, we observe a Kramers' nondegeneracy: the existence of a nondegenerate eigenstate of any time reversal symmetric n-qubit Hamiltonian demands (i) n even and (ii) maximal concurrence of said eigenstate. We provide examples of how to apply this work to study the kinematics and dynamics of entanglement in spin chain Hamiltonians.
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"abstract": "For n an even number of qubits and v a unitary evolution, a matrix\ndecomposition v=k1 a k2 of the unitary group is explicitly computable and\nallows for study of the dynamics of the concurrence entanglement monotone. The\nside factors k1 and k2 of this Concurrence Canonical Decomposition (CCD) are\nconcurrence symmetries, so the dynamics reduce to consideration of the a\nfactor. In this work, we provide an explicit numerical algorithm computing v=k1\na k2 for n odd. Further, in the odd case we lift the monotone to a two-argument\nfunction, allowing for a theory of concurrence dynamics in odd qubits. The\ngeneralization may also be studied using the CCD, leading again to maximal\nconcurrence capacity for most unitaries. The key technique is to consider the\nspin-flip as a time reversal symmetry operator in Wigner\u0027s axiomatization; the\noriginal CCD derivation may be restated entirely in terms of this time\nreversal. En route, we observe a Kramers\u0027 nondegeneracy: the existence of a\nnondegenerate eigenstate of any time reversal symmetric n-qubit Hamiltonian\ndemands (i) n even and (ii) maximal concurrence of said eigenstate. We provide\nexamples of how to apply this work to study the kinematics and dynamics of\nentanglement in spin chain Hamiltonians.",
"arxiv_id": "quant-ph/0402051",
"authors": [
"Stephen S. Bullock",
"Gavin K. Brennen",
"Dianne P. O\u0027Leary"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.1900293",
"journal_ref": "Journal of Mathematical Physics, volume 46, 062104 (2005)",
"title": "Time Reversal and n-qubit Canonical Decompositions",
"url": "https://arxiv.org/abs/quant-ph/0402051"
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