1. Fundamental concepts

我主要是在搞量子信息方面的东西，因此对这部分有所侧重，量子计算就直接跳过了。不过我们先从本书的量子力学开始看起。

2 Infroduction to quantum menchanics

2.1.8 Operator functions

unintary similarity transformation: ​A\rightarrow UAU^{\dagger}, as ​tr(UAU^{\dagger})=tr(U^{\dagger}UA)=tr(A).

  • ​tr(A|\psi\rangle\langle\psi|)=\langle\psi|A|\psi\rangle.

2.2 The postulates of quantum mechanics

2.2.1 State space

2.2.2 Evolution

幺正算子是保内积的，模也不变，也就是说变换前后没有粒子的产生和粒子的湮灭，它的概率也是守恒的，初始时量子态是归一的，那么演化后也是归一的。如果说厄米算子是表示物理量，那么幺正算子就是在表示时间演化。

How does the state ​|\psi\rangle change with time?
The evolution of a closed quantum system is described by a unitary transformation.
That is

|\psi^{\prime}\rangle=U|\psi\rangle

some unitary operators: paili matrices, Hadamard gate.
A more refined version of this postulate:

i \hbar\frac{d|\psi\rangle}{dt}=H|\psi\rangle

​H is a fixed Hermitian operator known as the Hamiltonian of the closed system.
What is the connection between the Hamiltonian and the unitary operator:

|\psi(t_2)\rangle=\exp\left[\frac{-iH(t_2-t_1)}{\hbar}\right]|\psi(t_1)\rangle=U(t_1,t_2)|\psi(t_1)\rangle

2.2.3 Quantum measurement

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. If the state of the quantum system is ​|\psi\rangle immediately before the measurement then the probability that result ​m occurs is given by

p(m)=\langle\psi|M_m^\dagger M_m|\psi\rangle

and the state of the system after the measurement is

\frac{M_m|\psi\rangle}{\sqrt{\langle\psi|M_m^\dagger M_m|\psi\rangle}}.

The measurement operators satisfy the completeness equation,

\sum_mM_m^\dagger M_m=I

The completeness equation expresses the fact that probabilities sum to one:

1=\sum_mp(m)=\sum_m\langle\psi|M_m^\dagger M_m|\psi\rangle.

顺便一提这里，我思维卡了一下对上面这个式子感到迷惑，但实际上应该就是​\langle \psi_i|I|\psi_j\rangle=\delta_{ij}=1.顺便一提上面这个测量叫做一般测量.还有一些其他作为分支单独提出的特殊测量比如投影测量（冯·诺依曼测量）和POVM测量等.

2.2.4 Distinguishing quantum states

non-orthogonal quantum states cannot be distinguished.

2.2.5 Projective measurements

投影测量，说曹操曹操到。
Indeed, projective measurements actually turn out to be equivalent to the general measurement postulate, when they are augmented with the ability to perform unitary transformations.
Projective measurements: A projective measurement is described by an observable, ​M , a Hermitian operator on the state space of the system being observed. The observable has a spectral decomposition,

M=\sum_mmP_m,

where ​P_m is the projector onto the eigenspace of ​M with eigenvalue ​m. The possible outcomes of the measurement correspond to the eigenvalues, ​m, of the observable. Upon measuring the state ​|\psi\rangle, the probability of getting result ​m is given by

p(m)=\left\langle\psi|P_m|\psi\right\rangle.

Given that outcome m occurred, the state of the quantum system immediately after the measurement is

\frac{P_m|\psi\rangle}{\sqrt{p(m)}}.

当​M_m为厄米算子并且满足​M_mM_{m^{\prime}}=\delta_{m,m^{\prime}}M_m的时候，2.2.3中提到的一般测量就可以变为投影测量。
Projective measurements have many nice propertie, the average value of the measurement is

\begin{aligned} \mathbf{E}(M)& =\sum_mmp(m) \\ &=\sum_mm\langle\psi|P_m|\psi\rangle \\ &=\langle\psi|\left(\sum_mmP_m\right)|\psi\rangle \\ &=\langle\psi|M|\psi\rangle. \end{aligned}

From this formula for the average follows a formula for the standard deviation associated to observations of ​M,

\begin{aligned}(\Delta(M))^{2}&=\langle(M-\langle M\rangle)^2\rangle\\&=\langle M^2\rangle-\langle M\rangle^2.\end{aligned}

但是搞不懂实际要怎么算，汗流浃背了。待会听听课看看是怎么个事。

这边也再补充一点吧，更详细的推导，可能符号和前面用的不太一样凑合看看，

\begin{aligned} \langle\hat{\Omega}\rangle & \equiv\sum_ip_ie_i=\sum_i\left|c_i\right|^2e_i \\ &=\sum_i\left|\langle e_i|\psi\rangle\right|^2e_i \\ &=\sum_i\left\langle\psi|\boldsymbol{e}_i\right\rangle\left\langle\boldsymbol{e}_i|\psi\right\rangle\boldsymbol{e}_i \\ &=\langle\psi|(\sum_ie_i|e_i\rangle\langle e_i|)|\psi\rangle \\ &=\langle\psi|\hat{\Omega}|\psi\rangle \\ &=\mathrm{Tr}(\hat{\Omega}|\psi\rangle\langle\psi|) \end{aligned}

其实是计算了一个期望值。
投影算子又称密度算子。

2.2.6 POVM measurements

这种测量不关心测量后的状态，而关心各个测量结果的概率。

2.4 The density operator

2.4.1 Ensembles of quantum states

Suppose a quantum system is in one of a number of states ​|\psi_i\rangle, where ​i is an index, with respective probabilities ​p_i. We shall call ​\{p_i, |\psi_i\rangle\} an ensemble of pure states. The density operator for the system is defined by the equation

\rho\equiv\sum_ip_i|\psi_i\rangle\langle\psi_i|.

密度算子也可以描述所有的量子力学假设，我们可以针对不同情况从向量语言和密度算子语言中找到更加好用的去使用。
example: the evolution of the density operator is described by the equation

\rho=\sum_ip_i|\psi_i\rangle\langle\psi_i|\xrightarrow{U}\sum_ip_iU|\psi_i\rangle\langle\psi_i|U^\dagger=U\rho U^\dagger.

and there are also easily described in the density operator languang.

p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle=\mathrm{tr}(M_m^\dagger M_m|\psi_i\rangle\langle\psi_i|)

the probability of obtaining result m is

\begin{aligned} p(m)& =\sum_ip(m|i)p_i \\ &=\sum_ip_i\mathrm{tr}(M_m^\dagger M_m|\psi_i\rangle\langle\psi_i|) \\ &=\mathrm{tr}(M_m^\dagger M_m\rho). \end{aligned}

得到结果 ​m 后，态就会变为

|\psi_i^m\rangle=\frac{M_m|\psi_i\rangle}{\sqrt{\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle}}.

以上解释可以参考一下投影算符，那么投影算符为啥长那样？或许……因为是公设？
此时对于​|\psi_i^m\rangle这个态的集合，或者叫他系综，相应的密度算子为

\rho_m=\sum_ip(i|m)|\psi_i^m\rangle\langle\psi_i^m|=\sum_ip(i|m)\frac{M_m|\psi_i\rangle\langle\psi_i|M_m^\dagger}{\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle}.

由概率论小知识​\begin{aligned}p(i|m)=p(m,i)/p(m)=p(m|i)p_i/p(m)\end{aligned}我们可以进一步得到

\begin{aligned} \rho_{m}& =\sum_ip_i\frac{M_m|\psi_i\rangle\langle\psi_i|M_m^\dagger}{\mathrm{tr}(M_m^\dagger M_m\rho)} \\ &=\frac{M_m\rho M_m^\dagger}{\mathrm{tr}(M_m^\dagger M_m\rho)}. \end{aligned}

A quantum system whose state ​|\psi\rangle is known exactly is said to be in a pure state. In this case the density operator is simply ​\rho=|\psi\rangle\langle\psi|. Otherwise, ​\rho is in a mixed state; it is said to be a mixture of the different pure states in the ensemble for ​\rho.
pure state satisfies ​tr(\rho^2 ) = 1, while a mixed state satisfies ​tr(\rho^2 ) < 1.
imagine a quantum system is prepared in the state ​\rho_i with probability ​p_i, the system may be desceibed by the density matrix ​\sum\limits_i p_i\rho_i.

2.4.2 General properties of the density operator

Throerm: (Characterization of density operators) An operator ​\rho is the density
operator associated to some ensemble ​\{p_i, |\psi_i\rangle\} if and only if it satisfies the
conditions:

  1. (Trace condition) ​\rho has trace equal to one.
  2. (Positivity condition) ​\rho is a positive operator.
    密度算子是厄米的，半正定的，且迹为1。
    接下来就可以用密度矩阵改写量子力学四大公设了：

  • Associated to any isolated physical system is a complex vector space with inner product (that is, a Hilbert space) known as the state space of the system. The system is completely described by its density operator, which is a positive operator ​\rho with trace one, acting on the state space of the system. If a quantum system is in the state ​\rho_i with probability ​p_i, then the density operator for the system is ​\sum\limits_ i p_i\rho_i.

  • The evolution of a closed quantum system is described by a unitary transformation. That is, the state ​\rho of the system at time ​t_1 is related to the state ​\rho^{\prime} of the system at time ​t_2 by a unitary operator ​U which depends only on the times ​t_1 and ​t_2, ​\rho^{\prime}=U\rho U^\dagger.

  • 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. If the state of the quantum system is ​\rho immediately before the measurement then the probability that result ​m occurs is given by

    p(m)=\mathrm{tr}(M_m^\dagger M_m\rho),

    and the state of the system after the measurement is

    \frac{M_m\rho M_m^\dagger}{\mathrm{tr}(M_m^\dagger M_m\rho)}.

    The measurement operators satisfy the completeness equation,

    \sum_mM_m^\dagger M_m=I.

  • The state space of a composite physical system is the tensor product of the state spaces of the component physical systems. Moreover, if we have systems numbered ​1 through ​n, and system number ​i is prepared in the state ​\rho_i, then the joint state of the total system is ​\rho_1\otimes\rho_2\otimes\ldots\rho_n.

不是哥们，我居然一次直积都没有提到过？？
在这里补充一下
For ​A=\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix} on ​\mathcal{H}_1 and ​B=\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix} on ​\mathcal{H}_2, the direct product operator on ​\mathcal{H}_1\otimes\mathcal{H}_2 is defined as

A\otimes B=\begin{pmatrix}a_{11}B&a_{12}B\\a_{21}B&a_{22}B\end{pmatrix}

这里有一个常见的误会，两种不同的量子态系综也可以产生相同的密度矩阵。
these two different ensembles of quantum states give rise to the same density matrix.
In general, the eigenvectors and eigenvalues of a density matrix just indicate one of many possible ensembles that may give rise to a specific density matrix, and there is no reason to suppose it is an especially privileged ensemble.
一般而言，密度矩阵的特征向量和特征值只是表示可能产生特定密度矩阵的众多可能系综中的一种，没有理由假设它是一种特别特权系综。
A natural question to ask in the light of this discussion is what class of ensembles does give rise to a particular density matrix?
根据这个讨论，一个自然的问题是什么样的系综会产生一个特定的密度矩阵?
​|\tilde{\psi}_i\rangle， which may not be normalized to unit length. We say the set ​|\tilde{\psi}_i\rangle generates the operator ​\rho\equiv\sum_i|\tilde{\psi}_i\rangle\langle\tilde{\psi}_i|, and thus the connection to the usual ensemble picture of density operators is expressed by the equation ​|\tilde{\psi}_i\rangle=\sqrt{p_i}|\psi_i\rangle.
Theorem 2.6:(Unitary freedom in the ensemble for density matrices)Thesets ​|\tilde{\psi}_i\rangle and ​|\tilde{\varphi}_i\rangle generate the same density matrix if and only if

|\tilde{\psi}_i\rangle=\sum_ju_{ij}|\tilde{\varphi}_j\rangle,

密度矩阵系综中的单位自由度
where ​u_{ij} is a unitary matrix of complex numbers, with indices ​i and ​j,and we ‘pad’ whichever set of vectors ​|\tilde{\psi}_i\rangle or ​|\tilde{\varphi}_i\rangle is smaller with additional vectors ​0 so that the two sets have the same number of elements.
作为定理的推论，
note that ​\rho=\sum_ip_i|\psi_i\rangle\langle\psi_i|=\sum_jq_j|\varphi_j\rangle\langle\varphi_j| for normalized states ​|\tilde{\psi}_i\rangle, ​|\tilde{\varphi}_i\rangle and probability distributions ​p_i and ​q_j if and only if

\sqrt{p_i}|\psi_i\rangle=\sum_ju_{ij}\sqrt{q_j}|\varphi_j\rangle,

此处应有证明，但是先略（目移

2.4.3 The reduced density operator

系统​A的约化密度算子定义
Suppose we have physical systems ​A and ​B, whose state is described by a density operator ​ρ^{AB}. The reduced density operator for system ​A is defined by

\rho^A\equiv\mathrm{tr}_B(\rho^{AB}),

来喽！偏迹！
式中​tr_B是系统​B上称为部分迹的算子的映射
where ​tr_B is a map of operators known as the partial trace over system ​B. The partial trace is defined by

\mathrm{tr}_B\left(|a_1\rangle\langle a_2|\otimes|b_1\rangle\langle b_2|\right)\equiv|a_1\rangle\langle a_2|\operatorname{tr}(|b_1\rangle\langle b_2|),

where ​|a_1\rangle and ​|a_2\rangle are any two vectors in the state space of ​A,and ​|b_1\rangle and ​|b_2\rangle are any two vectors in the state space of ​B. The trace operation appearing on the right hand side is the usual trace operation for system ​B, so ​\mathrm{tr}(|b_1\rangle\langle b_2|)=\langle b_2|b_1\rangle. We have defined the partial trace operation only on a special subclass of operators on ​AB.
The reduced density operator for system ​A is in any sense a description for the state of system ​A. The physical justification for making this identification is that the reduced density operator provides the correct measurement statistics for measurements made on system ​A.
系统​A的约化密度算符在任何意义上都是系统​A状态的一种描述。进行这种辨识的物理理由是，约化密度算符为在系统​A上进行的测量提供了正确的测量统计量。