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研究生:梁昶國
研究生(外文):Chang-KuoLiang
論文名稱:雙量子位元糾纏之分析
論文名稱(外文):Qubit-qubit Entanglement and Its Analysis
指導教授:許祖斌
指導教授(外文):Chopin Soo
學位類別:碩士
校院名稱:國立成功大學
系所名稱:物理學系碩博士班
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2010
畢業學年度:98
語文別:英文
論文頁數:23
外文關鍵詞:entanglementqubitchshvon NeumannWooterconcurrenceentanglement evolution
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吾人提出了雙量子位元純態之糾纏最重要的理論以及數學性質的回顧及解析。在簡明地敘述了希爾伯特空間的幾何之後;藉由詮釋雙量子位元純態之幾何以及透過奇異值分解與施密特數的研究,吾人推導出描述雙量子位元純態之糾纏的測度。對雙量子位元系統而言,該測度於幾何上我們可以視其為S^7/[S^3?S^3 ]的結構。吾人據此推導出的雙量子位元純態之糾纏測度被證明與許多著名的糾纏測度同調;例如約化密度矩陣的von Neumann entropy,CHSH算符之最大期望值,以及Wooter的concurrence。接著吾人給出針對糾纏測度之時間演化的完整描述,並且完整討論了糾纏測度於Schro ?dinger演化下的充分與必要條件,並且附上兩個有意義的範例作為說明。而Anandan-Aharonov 相位也被證明與雙量子位元純態之糾纏無關。
An analysis and review of the most important theoretical and mathematical properties of pure bipartite qubit-qubit entanglement is presented. After a short description of the geometry of Hilbert spaces, the entanglement measure is derived, both from a geometrical perspective and also by singular value decomposition and Schmidt number analysis. For bipartite qubit-qubit systems, the entanglement can be identified with the quotient space $[S^{7}/S^{3}otimes S^3]$. The measure of entanglement derived is shown to be monotonic to various other well-known entanglement parameters such as the von Neumann entropy of the reduced density matrix, maximum of the expectation value of Clauser-Horne-Shimony-Holt operator, and Wooters' concurrence. A complete description of the time evolution of the entanglement parameter is given. The necessary and sufficient condition for entanglement to be preserved under Schrodinger evolution is discussed, and illustrated explicitly with nontrivial examples. It is also confirmed that entanglement is independent of Abelian geometric phase or the Anandan-Aharonov phase.
1. Introduction and Overview 3
2. Geometry of Hilbert space 5
A. Spherical geometry of (N+1)-dimensional Hilbert Space 5
B. Geometrical Description of Bipartite Entanglement 5
3. Bipartite Qubit-Qubit Entanglement 6
A. Bell Basis 6
B. Entanglement parameter of bipartite qubit-qubit systems 7
C. Bell states and maximum entanglement 8
D. Other measures of entanglement 9
4. Evolution of Entanglement 13
5. Explicit Examples of Bipartite Qubit-Qubit Entanglement and Its Evolution 16
A. Time Independent Evolution of Entanglement 16
B. Periodic Time Evolution of Entanglement 18
6. Independence of Qubit-Qubit Bipartite Entanglement from Geometric Phase 22
References 23
[1] Discussion of probability relations between separated systems, Schr?odinger, Erwin. Proceeding of Cambridge Philosophical Society 31.555-563, 1935
[2] On the Einstein-Podolsky-Rosen paradox, J.S. Bell, Physics, Vol. 1, No. 3,195-200, 1964
[3] Proposed experiment to test Local Hidden Variable theories, J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23,880-884, 1969
[4] Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen chan- nels, Charles H. Bennett, Gilles Brassard, Claude Crepeau, Richard Jozsa, Asher Peres, and William K. Wootters Phys. Rev. Lett. 70, 1895-1899, 1993
[5] Entanglement of formation of an arbitrary state of two qubits, William K. Wootters, Phys. Rev. Lett. 80, 2245-2248, 1998
[6] Notes on the exact Geometric Phase and Geometrical Description of Entanglement, C.Soo, seminars at the Dept. of Physics, National University of Singapore, Febuary-April, 2008
[7] See for instance, Quantum Computation and Quantum Information, Michael A. Nielsen; Isaac L.Chuang. Cambridge University Press,New York, 2000
[8] Matrix Computations, Gene H Golub; Charles F Van Loan, Baltimore, Md. [u.a.] Johns Hopkins Univ. Press, 1996
[9] A Study of the Clauser-Horne-Shimony-Holt Relation, Neo Kai-Siang, M.Sc Thesis, National Cheng Kung University, 2005
[10] Wigner Rotations, Bell States, and Lorentz Invariance of Entanglement and von Neumann Entropy, C. Soo and C. Y. Lin, Int. J. Quantum Info.183-200, Febuary, 2004
[11] A Study of the Geometric Phase in Quantum Mechanics, Hui-Chen Lin, M.Sc Thesis, National Cheng Kung University, 2009
[12] Geometric Quantum Phase and Angles, J. Anandan and Y. Aharonov, Phys. Rev. D 38, 1863-1870, 1988
[13] Geometrical Description of Berry's Phase, Don N. Page Phys. Rev. A 36, 3479-3481, 1987
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