Geometry of two-qubit system and hopf fibration
Complex Hopf fibration and quaternionic Hopf fibration provide distinct ways of describing the state space for two-qubit quantum states. In this research, we have studied the geometry of quantum states for a two-level quantum system, along with the correspondence between complex Hopf bundle and q...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2013
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/67679/1/IPM%202013%2011%20IR.pdf |
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| Summary: | Complex Hopf fibration and quaternionic Hopf fibration provide distinct ways of
describing the state space for two-qubit quantum states. In this research, we have
studied the geometry of quantum states for a two-level quantum system, along with
the correspondence between complex Hopf bundle and quaternionic Hopf bundle.
In the first part of our study, we investigate the behaviour of local coordinates for
both Hopf bundles under different degree of entanglement such as entangled states
and non-entangled states. Fubini-Study metric for complex projective space is also
obtained. Its form suggests that for the intermediate entangled states, complex
projective space CP3 can be described as a set of flat three-tori parametrized by
a three-sphere. The local inhomogeneous coordinate of CP3 (base space of complex
Hopf fibration) is found to carry the description of both subsystems A and B,
whereas in the case of maximally entangled state, basis elements of local coordinates
CP3 is not linearly independent of each other. Next, we construct a base space map between CP3 and S4, which is denoted as η
map. After the mapping, we obtained phases in the base space manifold, different
sections and coordinate charts are related by transition functions. We found that
there is an inherent symmetry of coordinate transformation corresponds to different
sections of CP3, which is expressed in terms of transition functions having the U(1)
group structure. Also under η map, phases and transition function in S4 is doubled
over that of CP3, indicating subtle symmetric changes after the mapping. The base
space coordinates of quaternionic Hopf bundle are consist of two parts, whereby the
first part is invariant to the coordinate transformation in CP3 but sensitive to the
coordinate transformation in S4. |
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