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Geometry of mixed states for a q-bit and the quantum Fisher information tensor


Ercolessi, E; Schiavina, M (2012). Geometry of mixed states for a q-bit and the quantum Fisher information tensor. Journal of Physics A: Mathematical and General, 45(36):365303-17pp.

Abstract

After a review of the pure state case, we discuss from a geometrical point of view the meaning of the quantum Fisher metric in the case of mixed states for a two-level system, i.e. for a q-bit, by examining the structure of the fiber bundle of states, whose base space can be identified with a co-adjoint orbit of the unitary group. We show that the Fisher information metric coincides with the one induced by the metric of the manifold of SU(2), i.e. the three-dimensional sphere S-3, when the mixing coefficients are varied. We define the notion of Fisher tensor and show that its anti-symmetric part is intrinsically related to the Kostant-Kirillov-Souriau symplectic form that is naturally defined on co-adjoint orbits, while the symmetric part is non-trivially again represented by the Fubini-Study metric on the space of mixed states, weighted by the mixing coefficients.

Abstract

After a review of the pure state case, we discuss from a geometrical point of view the meaning of the quantum Fisher metric in the case of mixed states for a two-level system, i.e. for a q-bit, by examining the structure of the fiber bundle of states, whose base space can be identified with a co-adjoint orbit of the unitary group. We show that the Fisher information metric coincides with the one induced by the metric of the manifold of SU(2), i.e. the three-dimensional sphere S-3, when the mixing coefficients are varied. We define the notion of Fisher tensor and show that its anti-symmetric part is intrinsically related to the Kostant-Kirillov-Souriau symplectic form that is naturally defined on co-adjoint orbits, while the symmetric part is non-trivially again represented by the Fubini-Study metric on the space of mixed states, weighted by the mixing coefficients.

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Additional indexing

Item Type:Journal Article, not refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:14 September 2012
Deposited On:23 Jan 2013 14:27
Last Modified:05 Apr 2016 16:20
Publisher:IOP Publishing
ISSN:0305-4470
Publisher DOI:https://doi.org/10.1088/1751-8113/45/36/365303

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