3. Quantum information

quantum noise and quantum operations

This is a central topic of the third part of this book, which begins in this chapter with the description of the quantum operations formalism, a powerful set of tools enabling us to describe quantum noise and the behavior of open quantum systems.
我们需要研究的是一个开放的量子系统，这个系统可以和外界产生相互作用，也因此会产生噪声。我们从描述量子运算形式主义开始。

8.1 Classical noise and Markov processes

首先来看看经典系统中的噪声描述：Imagine a bit is being stored on a hard disk drive attached to an ordinary classical computer. The bit starts out in the state ​0 or ​1, but after a long time it becomes likely that stray magnetic fields will cause the bit to become scrambled, possibly flipping its state. We can model this by a probability ​p for the bit to flip, and a probability ​1 − p for the bit to remain the same.
Markov process: 所谓马尔可夫过程，我的理解其实就是对于两个门来说，后面的门坏没坏和前面的门没有关系，就称之为马尔可夫过程。
对于一个阶段的过程，可以简单写作​\vec q=E\vec p, 这当中 ​E 成为概率转移矩阵(evolution matrix)。
当然 ​E 也是有要求的:(positivity)所有entries是非负的，(completeness)每一列的和为1。

8.2 Quantum operations

8.2.1 overview

  • physically motivated aximos
  • operator-sum representation
  • system coupled to environment

8.2.2 Environments and quantum operations

我们来描述一个开放的量子力学系统，然而只有封闭的系统才能只用酉变换来描述，所以我们最终得到的状态一般不能用酉变换表示，而是用这个：

\mathcal{E}(\rho)=\mathrm{tr}_\mathrm{env}\left[U\left(\rho\otimes\rho_\mathrm{env}\right)U^\dagger\right].

可以看出最终状态是由主系统(principal)和环境(environment)相互作用产生的，他们共同构成了一个封闭的量子系统。

An important assumption is made in this definition – we assume that the system and the environment start in a product state. In general, of course, this is not true.
假设是从product的态开始的，但是这不正确。然而我没有看懂为什么说是不正确的（。
Even if this is not the case, we shall see later that the quantum operations formalism can even describe quantum dynamics when the system and environment do not start out in a product state.
然而作者又说即使不是，稍后也会看到这个假设无关紧要，量子运算形式还是可以描述量子动力学。拭目以待。
接下来，有一个重要的问题，那就是​U 要如何描述一个无限自由度的系统？
如果主系统是一个​d维希尔伯特空间，那么环境就是一个不超过​d^2的希尔伯特空间。环境一开始不需要为混合态，纯态也是足够的。
我们将量子操作描述为由主要系统和环境相互作用产生，不过我们可以进行一点点概念扩展：

\mathcal{E}(\rho)=\rho'=\mathrm{tr}_A(U(\rho\otimes|0\rangle\langle0|)U^\dagger).

处于未知state ​\rho 的单个量子位记为系统A，在标准态​|0\rangle 的系统记为B。相互作用后丢弃A，B为最终态​\rho^{\prime}。
some initial system is prepared in an unknown quantum state ​\rho, and then brought into contact with other systems prepared in standard states, allowed to interact according to some unitary interaction, and then some part of the combined system is discarded, leaving just a final system in some state ​\rho^{\prime}. The quantum operation ​\mathcal{E} defining this process simply maps ​\rho to ​\rho^{\prime}.
我对此的理解就是我们不再需要主系统和环境的区分了，他们都叫一个系统。

8.2.3 Operator-sum representation

\begin{aligned}\mathcal{E}(\rho)&=\sum_k\langle e_k|U\left[\rho\otimes|e_0\rangle\langle e_0|\right]U^\dagger|e_k\rangle\\&=\sum_kE_k\rho E_k^\dagger,\end{aligned}

where ​E_k\equiv\langle e_k|U|e_0\rangle is an operator on the state space of the principal system.
The operators ​{E_k} are known as operation elements for the quantum operation ​\mathcal{E}.
这里操作元也是有完备性要求的，即：

\begin{aligned} \text{1}& =\mathrm{tr}(\mathcal{E}(\rho)) \\ &=\mathrm{tr}\left(\sum_kE_k\rho E_k^\dagger\right) \\ &=\mathrm{tr}\left(\sum_kE_k^\dagger E_k\rho\right). \end{aligned}

他要求迹为1，因此我们必须有：

\sum_kE_k^\dagger E_k=I.

这是一个保迹的等式。
对比一下我之前写过的Shankar的量子力学笔记中的数学描述这一章里对于矩阵元的介绍，是不是很眼熟。
当然我们也可以不去保迹，只要通过测量获取额外的信息来弥补就好。
算子和这一表示方法可以让我们忽略环境的属性。同时我们需要探讨几个问题。首先对任何算子和以及开放量子系统，他们之间如何相互构建？

Physical interpretation of the operator-sum representation

Imagine that a measurement of the environment is performed in the basis ​|e_k\rangle after the unitary transformation ​U has been applied. Applying the principle of implicit measurement, we see that such a measurement affects only the state of the environment, and does not change the state of the principal system.
应用酉变换，即进行测量。应用隐式测量原理，即只改变环境状态，不改变系统状态。
Principle of implicit measurement: Without loss of generality, any unterminated quantum wires (qubits which are not measured) at the end of a quantum circuit may be assumed to be measured.
这是隐式测量准则，在量子计算这一章提到的。没太看懂（。
其中​\rho_k为主系统在结果​k发生时的状态

\begin{aligned} \rho_{k}\propto\mathrm{tr}_{E}(|e_{k}\rangle\langle e_{k}|U(\rho\otimes|e_{0}\rangle\langle e_{0}|)U^{\dagger}|e_{k}\rangle\langle e_{k}|)& =\langle e_{k}|U(\rho\otimes|e_{0}\rangle\langle e_{0}|)U^{\dagger}|e_{k}\rangle \\ &=E_{k}\rho E_{k}^{\dagger}. \end{aligned}

normalizing一下

\rho_k=\frac{E_k\rho E_k^\dagger}{\mathrm{tr}(E_k\rho E_k^\dagger)},

则结果​k的概率为

\begin{aligned} p(k)& =\mathrm{tr}(|e_k\rangle\langle e_k|U(\rho\otimes|e_0\rangle\langle e_0|)U^\dagger|e_k\rangle\langle e_k|) \\ &=\operatorname{tr}(E_k\rho E_k^\dagger). \end{aligned}

因此

\mathcal{E}(\rho)=\sum_kp(k)\rho_k=\sum_kE_k\rho E_k^\dagger.

举个栗子来解释量子和这种表示方法，我们选择​|e_{k}\rangle=|0_{E}\rangle和​|1_{E}\rangle 这个态，并且使用下标​P表示主系统，下标​E表示环境，则受控非门可以展开为

U=|0_P0_E\rangle\langle0_P0_E|+|0_P1_E\rangle\langle0_P1_E|+|1_P1_E\rangle\langle1_P0_E|+|1_P0_E\rangle\langle1_P1_E|.

因此

E_{0}=\langle0_E|U|0_E\rangle=|0_P\rangle\langle0_P|
E_{1}=\langle1_E|U|0_E\rangle=|1_P\rangle\langle1_P|,

因此

\mathcal{E}(\rho)=E_0\rho E_0+E_1\rho E_1,

这一部分就解决了开放系统到算子和表示的问题。不过这两个因为我没有看懂（

Measurements and the operator-sum representation

这一章是测量的算子和表示，为后续内容打基础
首先回忆一下从开放系统到算子和表示：
We have already found one answer: given the unitary system–environment transformation operation ​U , and a basis of states ​|e_k\rangle for the environment, the operation elements are

E_k\equiv\langle e_k|U|e_0\rangle.

It is possible to extend this result even further by allowing the possibility that a measurement is performed on the combined system–environment after the unitary interaction, allowing the acquisition of information about the quantum state. It turns out that this physical possibility is naturally connected to non-trace-preserving quantum operations, that is, maps​\mathcal{E}(\rho)=\sum_kE_k\rho E_k^\dagger such that ​\sum_kE_k^\dagger E_k\leq I.
这一段大概是在说根据开发系统的量子和表示进一步推导，进行测量可以有获得更多量子态的相关信息的可能性，而且这个测量应该是非保迹的。
我们接着给出如下阐述：首先设Q为主系统，E为环境，​\rho为主系统初始状态，​\sigma为环境初始状态，相互作用U是酉变换，则联合状态为

\rho^{QE}=\rho\otimes\sigma.

进行投影测量​P_m，没有测量的情况为The case where no measurement is made corresponds to the special case where there is only a single measurement outcome, ​m=0, which corresponds to the projector ​P_0\equiv I.
QE的最终状态为，

\frac{P_mU(\rho\otimes\sigma)U^\dagger P_m}{\mathrm{tr}(P_mU(\rho\otimes\sigma)U^\dagger P_m)},

变换和测量一锅乱烩。
我们目标是确定Q的最终状态作为初始状态的函数。假设测量结果m发生了，求出E，我们看到单独Q的最终状态是

\frac{\mathrm{tr}_E(P_mU(\rho\otimes\sigma)U^\dagger P_m)}{\mathrm{tr}(P_mU(\rho\otimes\sigma)U^\dagger P_m)}.

定义一个映射

\mathcal{E}_m(\rho)\equiv\mathrm{tr}_E(P_mU(\rho\otimes\sigma)U^\dagger P_m),

Q也是

\mathcal{E}_m(\rho)/\mathrm{tr}(\mathcal{E}_m(\rho)).

注意这里​\mathrm{tr}(\mathcal{E}_m(\rho)) 是测量结果m发生的概率。把上面定义的映射展开，得到

\begin{aligned} \mathcal{E}_m(\rho)& =\sum_{jk}q_{j}\mathrm{tr}_{E}(|e_{k}\rangle\langle e_{k}|P_{m}U(\rho\otimes|j\rangle\langle j|)U^{\dagger}P_{m}|e_{k}\rangle\langle e_{k}|) \\ &=\sum_{jk}E_{jk}\rho E_{jk}^{\dagger}, \end{aligned}

The quantum operations ​\mathcal{E}_m can be thought of as defining a kind of measurement process
generalizing the description of measurements given in Chapter 2.
​\mathcal{E}_m可以看作是定义了一种测量过程，是对第二章关于测量的描述的推广。

System–environment models for any operator-sum representation

这一章是任何运算和表示的系统环境模型，是系统到运算和的表示，对应最开始我们提到的问题。
Given a set of operators ​\{E_k\} is there some reasonable model environmental system and dynamics which give rise to a quantum operation with those operation elements?
In particular, we show that for any trace-preserving or non-trace-preserving quantum operation, ​\mathcal{E}, with operation elements ​\{E_k\}, there exists a model environment, ​E, starting in a pure state ​|e_0\rangle, and model dynamics specified by a unitary operator ​U and projector ​P onto ​E such that

\mathcal{E}(\rho)=\mathrm{tr}_E(PU(\rho\otimes|e_0\rangle\langle e_0|)U^\dagger P).

我们证明，对于任何具有运算元素 ​\{E_k\} 的保迹或非保迹量子操作 ​\mathcal{E}，都存在一个模型环境 ​E，从纯状态 ​|e_0\rangle 开始，并且模型由酉算子 ​U 和投影仪 ​P 指定到 ​E 上的动力学，使得有下面那一串公式。

具体是为什么呢？
To see this, suppose first that ​E is a trace-preserving quantum operation, with operator-sum representation generated by operation elements ​\{E_k\} satisfying the completeness relation ​\sum_{k}E_{k}^{\dagger}E_{k}=I, so we are only attempting to find an appropriate unitary operator ​U to model the dynamics. Let ​|e_k\rangle be an orthonormal basis set for ​E, in one-to-one correspondence with the index ​k for the operators ​E_k. Note that by definition ​E has such a basis; we are trying to find a model environment giving rise to a dynamics described by the operation elements ​\{E_k\}. Define an operator ​U which has the following action on states of the form ​|\psi\rangle|e_0\rangle.
首先，我们先整一个保迹的量子运算​\mathcal{E}，这是满足完备性关系的，所以我们只需要找到一个合适的幺正算子​U来建立动力学模型。定义一个​U，它对​|\psi\rangle|e_0\rangle有如下操作

U|\psi\rangle|e_0\rangle\equiv\sum_kE_k|\psi\rangle|e_k\rangle,

Note that for arbitrary states​|\psi\rangle and ​|\varphi\rangle of the principal system,

\langle\psi|\langle e_0|U^\dagger U|\varphi\rangle|e_0\rangle=\sum_k\langle\psi|E_k^\dagger E_k|\varphi\rangle=\langle\psi|\varphi\rangle,

啊嘞看起来是取了转置共轭但是​\langle\psi|\langle e_0|没有交换位置欸是没有影响的吗……
by the completeness relation. Thus the operator ​U can be extended to a unitary operator acting on the entire state space of the joint system. It is easy to verify that

\mathrm{tr}_E(U(\rho\otimes|e_0\rangle\langle e_0|)U^\dagger)=\sum_kE_k\rho E_k^\dagger,

so this model provides a realization of the quantum operation ​\mathcal{E} with operation elements ​\{E_k\}.
我们可以通过​\{E_k\}实现​\mathcal{E}了。

请看box
Mocking up a quantum operation
模拟量子运算
我们希望​U满足

E_k=\langle e_k|U|e_0\rangle,

where U is some unitary operator, and ​|e_k\rangle are orthonormal basis vectors for the environment system. Such a ​U is conveniently represented as the block matrix.
Note that the operatio elements ​E_k only determine the first block column of this matrix. Determination of the rest of the matrix is left up to us; we simply choose the entries such that ​U is unitary.

U=\begin{bmatrix} [E_1]&\cdot&&\cdot&&\cdot&&\cdots\\ [E_2]&\cdot&&\cdot&&\cdot&&\cdots\\ [E_3]&\cdot&&\cdot&&\cdot&&\cdots\\ [E_4]&\cdot&&\cdot&&\cdot&&\cdots\\ \vdots&&\vdots&&\vdots&&\vdots \end{bmatrix}

这一结论可以运用于非保迹的情况。
A more interesting generalization of this construction is the case of a set of quantum operations ​\{\mathcal{E}_{m}\} corresponding to possible outcomes from a measurement, so the quantum operation ​\sum_m\mathcal{E}_m is trace-preserving, since the probabilities of the distinct outcomes sum to one, ​1=\sum_mp(m)=\mathrm{tr}\left[\left(\sum_m\mathcal{E}_m\right)(\rho)\right] for all possible inputs ​\rho.
没太看明白这一段的因果关系。因为什么所以就保迹了？？

8.2.4 Axiomatic approach to quantum operations

量子运算的公理化方法
忘记之前学的东西，从公理化的角度入手。