dorsal/arxiv
View SchemaQuantum Channels and Representation Theory
| Authors | William Gordon Ritter |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0502153 |
| URL | https://arxiv.org/abs/quant-ph/0502153 |
| DOI | 10.1063/1.1945768 |
| Journal | J. Math. Phys. 46, 082103 (2005) |
Abstract
In the study of d-dimensional quantum channels $(d \geq 2)$, an assumption which is not very restrictive, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of SU(n) is a depolarizing channel for all $n$, but for most other representations this is not the case. Since the Bloch sphere is not appropriate here, we develop technology which is a generalization of Bloch's technique. Our method works by representing the density matrix as a polynomial in symmetrized products of Lie algebra generators, with coefficients that are symmetric tensors. Using these tensor methods we prove eleven theorems, derive many explicit formulas and show other interesting properties of quantum channels in various dimensions, with various Lie symmetry algebras. We also derive numerical estimates on the size of a generalized ``Bloch sphere'' for certain channels. There remain many open questions which are indicated at various points through the paper.
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"abstract": "In the study of d-dimensional quantum channels $(d \\geq 2)$, an assumption\nwhich is not very restrictive, and which has a natural physical interpretation,\nis that the corresponding Kraus operators form a representation of a Lie\nalgebra. Physically, this is a symmetry algebra for the interaction\nHamiltonian. This paper begins a systematic study of channels defined by\nrepresentations; the famous Werner-Holevo channel is one element of this\ninfinite class. We show that the channel derived from the defining\nrepresentation of SU(n) is a depolarizing channel for all $n$, but for most\nother representations this is not the case. Since the Bloch sphere is not\nappropriate here, we develop technology which is a generalization of Bloch\u0027s\ntechnique. Our method works by representing the density matrix as a polynomial\nin symmetrized products of Lie algebra generators, with coefficients that are\nsymmetric tensors. Using these tensor methods we prove eleven theorems, derive\nmany explicit formulas and show other interesting properties of quantum\nchannels in various dimensions, with various Lie symmetry algebras. We also\nderive numerical estimates on the size of a generalized ``Bloch sphere\u0027\u0027 for\ncertain channels. There remain many open questions which are indicated at\nvarious points through the paper.",
"arxiv_id": "quant-ph/0502153",
"authors": [
"William Gordon Ritter"
],
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"quant-ph"
],
"doi": "10.1063/1.1945768",
"journal_ref": "J. Math. Phys. 46, 082103 (2005)",
"title": "Quantum Channels and Representation Theory",
"url": "https://arxiv.org/abs/quant-ph/0502153"
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