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:	Wong, Wen Wei
Format:	Thesis
Language:	English
Published:	2013
Subjects:	Geometry algebraic Hopf algebras Algebra
Online Access:	http://psasir.upm.edu.my/id/eprint/67679/1/IPM%202013%2011%20IR.pdf
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spelling	my-upm-ir.676792019-03-22T03:27:15Z Geometry of two-qubit system and hopf fibration 2013-03 Wong, Wen Wei 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. Geometry, algebraic Hopf algebras Algebra 2013-03 Thesis http://psasir.upm.edu.my/id/eprint/67679/ http://psasir.upm.edu.my/id/eprint/67679/1/IPM%202013%2011%20IR.pdf text en public masters Universiti Putra Malaysia Geometry, algebraic Hopf algebras Algebra
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
topic	Geometry algebraic Hopf algebras Algebra
spellingShingle	Geometry algebraic Hopf algebras Algebra Wong, Wen Wei Geometry of two-qubit system and hopf fibration
description	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.
format	Thesis
qualification_level	Master's degree
author	Wong, Wen Wei
author_facet	Wong, Wen Wei
author_sort	Wong, Wen Wei
title	Geometry of two-qubit system and hopf fibration
title_short	Geometry of two-qubit system and hopf fibration
title_full	Geometry of two-qubit system and hopf fibration
title_fullStr	Geometry of two-qubit system and hopf fibration
title_full_unstemmed	Geometry of two-qubit system and hopf fibration
title_sort	geometry of two-qubit system and hopf fibration
granting_institution	Universiti Putra Malaysia
publishDate	2013
url	http://psasir.upm.edu.my/id/eprint/67679/1/IPM%202013%2011%20IR.pdf
_version_	1747812498515427328

Geometry of two-qubit system and hopf fibration

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