dorsal/arxiv
View SchemaOn Algebraic and Quantum Random Walks
| Authors | Demosthenes Ellinas |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0510128 |
| URL | https://arxiv.org/abs/quant-ph/0510128 |
Abstract
Algebraic random walks (ARW) and quantum mechanical random walks (QRW) are investigated and related. Based on minimal data provided by the underlying bialgebras of functions defined on e. g the real line R, the abelian finite group Z_N, and the canonical Heisenberg-Weyl algebra hw, and by introducing appropriate functionals on those algebras, examples of ARWs are constructed. These walks involve short and long range transition probabilities as in the case of R walk, bistochastic matrices as for the case of Z_N walk, or coherent state vectors as in the case of hw walk. The increase of classical entropy due to majorization order of those ARWs is shown, and further their corresponding evolution equations are obtained. Especially for the case of hw ARW, the diffusion limit of evolution equation leads to a quantum master equation for the density matrix of a boson system interacting with a bath of quantum oscillators prepared in squeezed vacuum state. A number of generalizations to other types of ARWs and some open problems are also stated. Next, QRWs are briefly presented together with some of their distinctive properties, such as their enhanced diffusion rates, and their behavior in respect to the relation of majorization to quantum entropy. Finally, the relation of ARWs to QRWs is investigated in terms of the theorem of unitary extension of completely positive trace preserving (CPTP) evolution maps by means of auxiliary vector spaces. It is applied to extend the CPTP step evolution map of a ARW for a quantum walker system into a unitary step evolution map for an associated QRW of a walker+quantum coin system. Examples and extensions are provided.
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"abstract": "Algebraic random walks (ARW) and quantum mechanical random walks (QRW) are\ninvestigated and related. Based on minimal data provided by the underlying\nbialgebras of functions defined on e. g the real line R, the abelian finite\ngroup Z_N, and the canonical Heisenberg-Weyl algebra hw, and by introducing\nappropriate functionals on those algebras, examples of ARWs are constructed.\nThese walks involve short and long range transition probabilities as in the\ncase of R walk, bistochastic matrices as for the case of Z_N walk, or coherent\nstate vectors as in the case of hw walk. The increase of classical entropy due\nto majorization order of those ARWs is shown, and further their corresponding\nevolution equations are obtained. Especially for the case of hw ARW, the\ndiffusion limit of evolution equation leads to a quantum master equation for\nthe density matrix of a boson system interacting with a bath of quantum\noscillators prepared in squeezed vacuum state. A number of generalizations to\nother types of ARWs and some open problems are also stated. Next, QRWs are\nbriefly presented together with some of their distinctive properties, such as\ntheir enhanced diffusion rates, and their behavior in respect to the relation\nof majorization to quantum entropy. Finally, the relation of ARWs to QRWs is\ninvestigated in terms of the theorem of unitary extension of completely\npositive trace preserving (CPTP) evolution maps by means of auxiliary vector\nspaces. It is applied to extend the CPTP step evolution map of a ARW for a\nquantum walker system into a unitary step evolution map for an associated QRW\nof a walker+quantum coin system. Examples and extensions are provided.",
"arxiv_id": "quant-ph/0510128",
"authors": [
"Demosthenes Ellinas"
],
"categories": [
"quant-ph"
],
"title": "On Algebraic and Quantum Random Walks",
"url": "https://arxiv.org/abs/quant-ph/0510128"
},
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