dorsal/arxiv
View SchemaMixed State Geometric Phase from Thomas Rotations
| Authors | Péter Lévay |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0312023 |
| URL | https://arxiv.org/abs/quant-ph/0312023 |
| DOI | 10.1088/0305-4470/37/16/009 |
| Journal | Journal of Physics A37 (2004) 4593-4605 |
Abstract
It is shown that Uhlmann's parallel transport of purifications along a path of mixed states represented by $2\times 2$ density matrices is just the path ordered product of Thomas rotations. These rotations are invariant under hyperbolic translations inside the Bloch sphere that can be regarded as the Poincar\'e ball model of hyperbolic geometry. A general expression for the mixed state geometric phase for an {\it arbitrary} geodesic triangle in terms of the Bures fidelities is derived. The formula gives back the solid angle result well-known from studies of the pure state geometric phase. It is also shown that this mixed state anholonomy can be reinterpreted as the pure state non-Abelian anholonomy of entangled states living in a suitable restriction of the quaternionic Hopf bundle. In this picture Uhlmann's parallel transport is just Pancharatnam transport of quaternionic spinors.
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"abstract": "It is shown that Uhlmann\u0027s parallel transport of purifications along a path\nof mixed states represented by $2\\times 2$ density matrices is just the path\nordered product of Thomas rotations. These rotations are invariant under\nhyperbolic translations inside the Bloch sphere that can be regarded as the\nPoincar\\\u0027e ball model of hyperbolic geometry. A general expression for the\nmixed state geometric phase for an {\\it arbitrary} geodesic triangle in terms\nof the Bures fidelities is derived. The formula gives back the solid angle\nresult well-known from studies of the pure state geometric phase. It is also\nshown that this mixed state anholonomy can be reinterpreted as the pure state\nnon-Abelian anholonomy of entangled states living in a suitable restriction of\nthe quaternionic Hopf bundle. In this picture Uhlmann\u0027s parallel transport is\njust Pancharatnam transport of quaternionic spinors.",
"arxiv_id": "quant-ph/0312023",
"authors": [
"P\u00e9ter L\u00e9vay"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/37/16/009",
"journal_ref": "Journal of Physics A37 (2004) 4593-4605",
"title": "Mixed State Geometric Phase from Thomas Rotations",
"url": "https://arxiv.org/abs/quant-ph/0312023"
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