Geometry of mixed states for a q-bit and the quantum Fisher information tensor
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 \mathsf {SU}(2), i.e. the three-dimensional sphere S3, 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.
- Publication:
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Journal of Physics A Mathematical General
- Pub Date:
- September 2012
- DOI:
- 10.1088/1751-8113/45/36/365303
- arXiv:
- arXiv:1205.2561
- Bibcode:
- 2012JPhA...45J5303E
- Keywords:
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- Quantum Physics;
- Mathematical Physics;
- 81S10 94A15
- E-Print:
- 20 pages