量子信息物理学2018chapt1_2_Kai Chen_print.ppt [兼容模式]

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1 量子信息物理学 PH 量子信息导论中国科学技术大学微尺度国家实验室 / 近代物理系陈凯

2 第一章量子体系量子态表示, 密度矩阵, 混合态, 量子不可克隆定理, Schmidt 分解, 量子测量等

3 任意两体纯态 Schmidt decomposition 假设 n m 存在一组正交态 µ 1, µ 2,, µ n 以及 ϕ 1, ϕ 2,, ϕ n 使得 ϕ 1, ϕ 2,, ϕ n 为的本征态参考 QCQI 2.5, M.A. Nielsen and I.L. Chuang

4 密度矩阵矩阵元的表示也即,,,

5 两比特密度矩阵用到

6 密度矩阵性质因此混合程度, 最小值? 纯态混合态

7 密度矩阵性质所有 N 维 Hilbert 空间 H 中的密度矩阵构成一个凸集一般地仍然为一个密度矩阵

8 密度矩阵演化作用一个幺正变换 U 在初态, 我们有密度矩阵变为

9 密度矩阵演化作用一个幺正变换 U 在一个系综上们有, 我密度矩阵变为

10 系综分解任意系综生成和表示正交归一基矢下的系综分解可以证明, 同一密度矩阵的两组系综之间存在一个幺正变换关系

11 Trace 操作对于混合态意味着对于各种量子态的测量输出总的概率为 1

12 Trace 操作性质矩阵对角元求和特性正交基展开

13 Trace 操作可分离态情形

14 Trace 操作纠缠态情形我们得到

15 Trace 操作一个例子我们得到量子态纯化 Purification

16 Trace 操作作用在矩阵上

17 约化密度矩阵 - 练习

18 量子测量沿基矢进行测量几率幅几率量子测量如对于任意叠加态

19 VonNeumann 测量 Von Neumann 测量是投影测量的一种类型给定一组正交基, 如果我们对于量子态实施沿基矢 Von Neumann 测量, 则有的

20 VonNeumann 测量例如考虑对于量子态实施相对于基矢测量的 Von Neumann 注意到因此我们有测得的几率为

21 VonNeumann 测量实际上

22 投影测量 (Projective measurements) 可观测量 M 存在一个谱分解满足获得结果 m 的几率为系统的量子态变为

23 投影测量 (Projective measurements) 可观测量 M 存在一个谱分解,, 更一般地定义可观测量称为自旋沿着轴方向的测量

24 广义测量 Quantum measurements are described by a collection {M m } of measurement operators. These are operators acting on the state space of the system being measured. The index m refers to the measurement outcomes that may occur in the experiment. 对于量子态, 结果 m 发生的几率为测量后的系统处于

25 广义测量其中测量算符满足完备性条件可见

26 例子我们有投影算子满足和从而同理测量之后量子态处于

27 广义测量是可实现的任意广义的量子测量均可以通过纠缠该系统与一个辅助系统, 应用一个幺正变换并实施投影测量来实现量子系统 A, 测量算符 M m, 辅助系统 B 定义可见实施投影测量从而有输出 m 的几率为

28 广义测量是可实现的任意广义的量子测量均可以通过纠缠该系统与一个辅助系统, 应用一个幺正变换并实施投影测量来实现量子系统 A, 测量算符 M m, 辅助系统 B 测量之后的联合量子态处于要测量的系统处于

29 量子系统 A, 测量算符 M m, POVM 测量 --PositiveOperator-ValuedMeasure 对于量子态, 结果 m 发生的几率为定义我们有可见集合 E m 本身就可以决定所有可能的不同测量输出的几率

30 How to distinguish non-orthogonal states optimally vs. Use generalized (POVM) quantum measurements. [see, e.g., Y. Sun, J. Bergou, and M. Hillery, Phys. Rev. A 66, (2002).] H-polarized photon 45 o -polarized photon The view from the laboratory: A measurement of a two-state system can only yield two possible results. If the measurement isn't guaranteed to succeed, there are three possible results: (1), (2), and ("I don't know"). Therefore, to discriminate between two non-orth. states, we need to use an expanded (3D or more) system. To distinguish 3 states, we need 4D or more. From Steinberg

31 The geometric picture 1 Θ 1 Θ 2 90 o From Steinberg Two non-orthogonal vectors The same vectors rotated so their projections onto x-y are orthogonal (The z-axis is inconclusive )

32 A test case Consider these three non-orthogonal states: Projective measurements can distinguish these states with certainty no more than 1/3 of the time. (No more than one member of an orthonormal basis is orthogonal to two of the above states, so only one pair may be ruled out.) But a unitary transformation in a 4D space produces: From Steinberg and these states can be distinguished with certainty up to 55% of the time

33 Experimental schematic (ancilla) From Steinberg

34 A 14-path interferometer for arbitrary 2-qubit unitaries... From Steinberg

35 Success! "Definitely 3" "Definitely 2" "Definitely 1" "I don't know" From Steinberg The correct state was identified 55% of the time-- Much better than the 33% maximum for standard measurements. M. Mohseni, A.M. Steinberg, and J. Bergou, Phys. Rev. Lett. 93, (2004)

36 参考 QCQI 2.2, M.A. Nielsen and I.L. Chuang M. Mohseni, A.M. Steinberg, and J. Bergou, Phys. Rev. Lett. 93, (2004)

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untitled Co-integration and VECM Yi-Nung Yang CYCU, Taiwan May, 2012 不列 1 Learning objectives Integrated variables Co-integration Vector Error correction model (VECM) Engle-Granger 2-step co-integration test Johansen