臺灣博碩士論文加值系統

Heisenberg uncertainty relation 在量子力學裡面扮演很重要的角色，但在這種類型的 uncertainty relation 並非適用於所有的情況。另一種常見的 uncertainty relation 是利用Shannon entropy 來定義所謂的entropic uncertainty。任意兩個 non-commuting observables 之間的entropic uncertainty, 其lower bound 可以藉由 Krauss 不等式得到良好的估計。然而, 任意三個 non-commuting observables 的 entropysum 卻沒有類似的不等式來估計 lower bound. 本論文藉由討論三個non-commuting spin operators Sx, Sy 以及 Sz 的entropy sum,來探究entropy sum 的極值和entropy sum 本身具有的對稱性如 mirror symmetries 或是 permutation symmetries 等等的關係.我們發現entropy sum 的極值很可能會發生在這些 symmetries 的 eigenstates上。此外我們也定義一種新型態的 information entropy, 並討論測量
這三個 spin operators 的 entropy sum 其極值發生時所對應的量子態.

Heisenberg uncertainty relation plays a very important role in the
quantum mechanics. However, the Heisenberg uncertainty has several
drawbacks and may not be a good description of uncertainty relation
among non-commuting observables. On the other hand the entropic
uncertainty that uses the Shannon entropy as its definition is widely
accepted as a measure of uncertainty in quantum mechanics. For any two
non-commuting observables we can have a good estimate on the lower
bound of the entropy sum by using the Krauss inequality. However, there
is no such inequality for more than two non-commuting observables. In
this thesis we discussed the entropy sum of the three non-commuting spin
operators Sx, Sy and Sz and calculated the maximum and the minimum
of the entropy sum. We found that the quantum states that give the
optimal values of the entropy sum are also eigenstates of some of the
symmetries, such as the mirror symmetries or the permutation symmetries,
of the entropy sum. In this thesis we also define a novel measure of
entropic uncertainty and then redo the calculation of the entropy sum for
the three spin observables.

目錄
中文摘要...........................................I
英文摘要...........................................II
誌謝..............................................III
目錄..............................................IV
圖目錄............................................VI
表目錄............................................VII
1. Introduction..................................1
1-1 Heisenberg uncertainty relation..............1
1-2 Entropic uncertainty.........................2
1-2.1: Joint spin measurement for spin-1/2 particle...............4
1-3 A new definition of Entropic uncertainty......................5
2. The Joint Spin Measurements....................................8
2-1 The Joint spin measurements and the entropic uncertainty......8
2-2 The symmetries of the entropic uncertainty....................9
2-2.1 The mirror symmetries.......................................9
2-2.2 The permutation symmetries..................................14
2-3 Numerical calculation of the entropic uncertainty.............17
2-3.1 Spin-1/2 pure states........................................17
2-3.2 Spin-1 pure states..........................................21
2-3.3 Spin-3/2 pure states........................................24
2-3.4 Spin-2 pure states..........................................27
3. The Joint Spin Measurements with the new entropic
uncertainty.......................................................31
3-1The new entropic uncertainty...................................31
3-2Numerical calculation of the new entropic uncertainty..........32
3-2.1 Spin-1/2 pure states........................................32
3-2.2 Spin-1 pure states..........................................36
4. Conclusion and discussion......................................39
References........................................................41
Appendix..........................................................42
圖目錄
圖2-1 鏡射圖.......................................................9
圖2-2 映射圖.......................................................10
圖2-3 Ixy 鏡射圖...................................................11
圖2-4 permutation transformation 空間作用圖.........................14
圖2-5 轉動軸在向量空間的位置..........................................19
圖2-6 state 在向量空間的位置圖........................................20
圖3-1 粒子通過3 個測量系統............................................31
表目錄
表(一) Ix,Iy 和Iz 作用spin eigenstate 的結果.........................12
表(二) Ixy 和Iyz 作用spin eigenstate 的結果..........................13
表(三) IyzIxy 作用spin eigenstate 的結果.............................15

[1]W. Heisenberg, Z. Phys. 43 , 172 (1927); H. P. Robertson, Phys. Rev.
34,163 (1929).
[2] D. Deutsch, Phys. Rev. Lett. 50 (1983) 631.
[3]M. H. Partovi, Phys. Rev. Lett. 50, 1883 (1983).
[4]I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129
(1975)
[5] H. Maassen and J.B.M. Uffink, Phys. Rev. Lett. 60, 1103 (1988).
[6]Jorge Sánchez, Phys. Lett. A 173 233-239 (1993)