dorsal/arxiv
View SchemaThe geometry of entanglement: metrics, connections and the geometric phase
| Authors | Peter Levay |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0306115 |
| URL | https://arxiv.org/abs/quant-ph/0306115 |
| DOI | 10.1088/0305-4470/37/5/024 |
| Journal | J.Phys.A37:1821-1842,2004 |
Abstract
Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.
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"abstract": "Using the natural connection equivalent to the SU(2) Yang-Mills instanton on\nthe quaternionic Hopf fibration of $S^7$ over the quaternionic projective space\n${\\bf HP}^1\\simeq S^4$ with an $SU(2)\\simeq S^3$ fiber the geometry of\nentanglement for two qubits is investigated. The relationship between base and\nfiber i.e. the twisting of the bundle corresponds to the entanglement of the\nqubits. The measure of entanglement can be related to the length of the\nshortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\\bf\nHP}^1$ between an arbitrary entangled state, and the separable state nearest to\nit. Using this result an interpretation of the standard Schmidt decomposition\nin geometric terms is given. Schmidt states are the nearest and furthest\nseparable ones lying on, or the ones obtained by parallel transport along the\ngeodesic passing through the entangled state. Some examples showing the\ncorrespondence between the anolonomy of the connection and entanglement via the\ngeometric phase is shown. Connections with important notions like the\nBures-metric, Uhlmann\u0027s connection, the hyperbolic structure for density\nmatrices and anholonomic quantum computation are also pointed out.",
"arxiv_id": "quant-ph/0306115",
"authors": [
"Peter Levay"
],
"categories": [
"quant-ph",
"hep-th"
],
"doi": "10.1088/0305-4470/37/5/024",
"journal_ref": "J.Phys.A37:1821-1842,2004",
"title": "The geometry of entanglement: metrics, connections and the geometric phase",
"url": "https://arxiv.org/abs/quant-ph/0306115"
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