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研究生：	吳青龍 Wu, Ching-Long
論文名稱：	維格納轉換和糾纏態的分析 Wigner transforms and analysis of entanglement
指導教授：	許祖斌 Soo, Chopin
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2005
畢業學年度：	93
語文別：	英文
論文頁數：	59
中文關鍵詞：	糾纏
外文關鍵詞：	Wigner transform, entanglement
相關次數：	點閱：57 下載：1
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　　本篇論文中介紹和位置、動量、自旋有關的維格納方程，然後我們主要考慮自旋部分的維格納方程，利用它來分析雙自旋的糾纏現象。我們也得到一個對密度矩陣分離性﹝”古典相關”或是”無糾纏”﹞的廣義必要條件。這個方法可以推廣到兩個自旋為n的狀況，而且也符合量子場論的要求和勞侖茲不變性。

　We study the Wigner function which is dependent on position, momentum, and spin; and apply the matrix of its spin part W, to the analysis of bipartite spin entanglement. We also derived a general necessary condition for separability (”classical correlation” or “non-entanglement”) of the density matrix. The method can be generalized to bipartite spin-n (or generic N = 2n + 1-state subsystems) and made compatible with the requirements of quantum field theory and Lorentz symmetry.

Introduction ......................................................1
1 Density matrix of mixed states of Bose and Fermi statistics .....1
2 Wigner Transform for spin-0 states ..............................2
3 Special case of two identical spin-0 particles ..................5
Wigner Transform for spin-1/2 system ..............................8
1 Introduction toWigner function for spin-1/2 states; and the
Clauser-Horne-Shimony-Holt (CHSH) relation ..........................8
1.1 The density matrix ............................................8
1.2 Wigner transforms and Quantum Field Theory (QFT) ..............9
1.3 Introduction to Wigner function for spin- 1/2 states ..........9
1.4 Introduction to (CHSH) .......................................11
2 Tr(WW^T) .......................................................12
2.1 General density matrix .......................................12
2.2 Density matrix of classically correlated system ..............13
2.3 Density matrix of pure states ................................13
2.4 Werner density matrix ........................................15
3 Tr(WW^TWW^T ) ..................................................16
3.1 General density matrix .......................................16
3.2 Density matrix of classically correlated system ..............17
3.3 Density matrix of pure states ................................18
3.4 Werner density matrix ........................................20
4 Tr(WW^TWW^TWW^T ) ..............................................21
4.1 General density matrix .......................................21
4.2 Density matrix of classically correlated system ..............27
4.3 Density matrix of pure states ................................30
4.4 Werner density matrix ........................................34
5 Eigenvalues of WW^T ............................................37
5.1 Cubic equation ...............................................38
5.2 Eigenvalues of WW^T for pure state system ....................38
5.3 Eigenvalues for classically correlated system ................40
5.4 Eigenvalues for Werner density matrix ........................40
6 Wigner transform and positivity of partial transpose (PPT) of
the density matrix .................................................42
6.1 Introduction to PPT ..........................................42
6.2 Wigner transforms and PPT ....................................42
6.3 PPT for Werner density matrix ................................43
6.4 Necessary condtion for classically correlated system in terns
of W ...............................................................43
6.5 Comparing Σ|λi|<=1 to PPT for Werner density matrix ........44
6.6 Explicit separability(decomposition) of Werner density matrix
(r <=1/3) ..........................................................45
Generalizing Wigner Transform to spin-n states ...................46
Conclusions ......................................................48
References .......................................................49
                                    

[1] E. P. Wigner, Physical Review, 40 (1932) 749-759; for a review, see also M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Distribution Functions in Physics:Fundamentals, Physics Reports 106(3) (1984) 121.
[2] C. Sean Bohun, Reinhard Illner, Paul F. Zweifel, and Le Matematiche Vol. XlVI (1991) Fasc.I, 429-438.
[3] R. F. O’Connell and E. P. Wigner, Physical Review A, 30 (1984), 2613-2618.
[4] B. S. Tsirelson. Lett. Math. Phys.(1980),4:93
[5] M.Horodecki, R. Horodecki, and R. Horodecki, Phy. Lett. A 223,1 (1996).
[6] T. D. Newton, E. P. Wigner, Rev. Mod. Phys. 21, 400 (1949).
[7] A. Sanpera, R.Tarrach, and G. Vidal, e-print quant-ph/9707041
[8] C. Soo and C. Y. Lin, International Journal of Quantum Information 2 (2004) 183-200
[9] Murray R. Spiegel, John Liu, Mathematical Handbook of Formulas and Tables Second Edition.
[10] C. Y. Lin, C. Soo and C. L. Wu,Wigner transforms and criteria for separability, in preparation (2005).
[11] R. F. Werner, Phys. Rev. A. 40, 4277 (1989)

校內：2006-07-04公開
校外：2006-07-04公開

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