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在《人类对人类的使用：控制论与社会》（1950 年）中，诺伯特·维纳以这种方式介绍了反馈控制：
“This control of a machine on the basis of its actual performance, rather than its expected performance is known as feedback . . It is the function of control . . . to produce a temporary and local reversal of the normal direction of entropy.”
“这种基于机器实际性能而非预期性能的控制被称为反馈……控制的功能……是产生熵的正常方向的暂时和局部逆转。”

The classic classroom example of feedback control is the mechanical governor used by James Watt in the eighteenth century to regulate the speed of his steam engines (Figure 31.1).
经典的课堂反馈控制示例是詹姆斯·瓦特在十八世纪使用的机械调速器，用于调节他的蒸汽机的速度（图 31.1）。

The actual engine speed raises the balls by centrifugal force. As these rise, the linkages are arranged to close down the intake valve. The speed is maintained in the neighborhood of an equilibrium.
实际发动机转速通过离心力抬起球体。当球体上升时，连杆被安排关闭进气阀。转速保持在平衡附近。
Feedback control was essential for the stable operation of steam engines, and was thus a critical enabling technology for these machines which powered the industrial revolution. A precise analysis was not made until the mid-1800s when Clerk Maxwell put his mind to it.
反馈控制对于蒸汽机的稳定运行至关重要，因此成为了推动工业革命的这些机器的关键技术。直到 19 世纪中叶，克拉克·麦克斯韦才对此进行了精确分析。

What is it that is so compelling about this apparatus? Firstly, it is easy to understand how it regulates the speed of a rotating steam engine. Secondly, and perhaps more importantly, it is a part of the device itself.
这个装置有什么如此吸引人的地方？首先，很容易理解它是如何调节旋转蒸汽机的速度的。其次，也许更重要的是，它是设备本身的一部分。
A naive observer would not distinguish this mechanical piece from the rest. And this device, if we think of it having a thought, almost needs no thoughts at all!
一个天真的观察者不会将这个机械部件与其他部件区分开来。而这个装置，如果我们认为它有思想，几乎不需要任何思想！
It need not know any real detail about the object it controls; no knowledge of steam, pressure, flow, friction, metal fatigue, anchor bolt placement, and so on. Almost nothing. Yet, it is the fundamental piece without which the steam engine might explode. Due
它不需要了解它控制的对象的任何真实细节；不需要了解蒸汽、压力、流量、摩擦、金属疲劳、锚栓位置等等。几乎什么都不需要。然而，它是蒸汽机的基本组成部分，没有它，蒸汽机可能会爆炸。

FIGURE 31.1 Boulton and Watt steam engine, 1788, showing the mechanical governor (metal ball mechanism) [located at London Science Museum].
图 31.1 博尔顿和瓦特蒸汽机，1788 年，显示机械调速器（金属球机制）[位于伦敦科学博物馆]。
to its seeming simplicity, the notion of feedback takes on a mysterious property. It is both intangible, and yet, fundamental to the stability of the device, because it responds to the effect of the actual rotational speed.
尽管反馈的概念看似简单，但它却具有神秘的特性。它既是无形的，又是设备稳定性的基础，因为它对实际转速的影响作出反应。
Without this simultaneously intangible and real feedback, the device would not exist!
没有这种同时无形又真实的反馈，这个设备就不存在！

Steam engines, are of course macroscopic systems described by classical physics, and control engineering has been founded on classical models. At this point in time, it is beginning to be possible to monitor and manipulate objects at the nanoscale.
蒸汽机当然是由经典物理描述的宏观系统，控制工程是建立在经典模型基础上的。此时，监测和操控纳米尺度的物体开始变得可能。
One can realistically contemplate controlling single atoms (Figure 31.2).
人们可以现实地考虑控制单个原子（图 31.2）。

At the atomic scale, the laws of quantum physics are needed, and in fact provide a significant new resource for technological exploitation, as can be seen in recent advances in quantum information and computing, precise metrology, atom lasers, quantum electromechanical systems, and quantum chemistry.
在原子尺度上，需要量子物理的法则，实际上为技术开发提供了重要的新资源，这在量子信息与计算、精密计量、原子激光、量子机电系统和量子化学的最新进展中得到了体现。
Quantum control refers to the control of physical systems whose behavior is dominated by the laws of quantum physics, and control theory is being developed that takes into account quantum physics (e.g., $[9-41]$$[9-41]$[9-41][9-41] ).
量子控制是指对物理系统的控制，这些系统的行为受量子物理法则的主导，正在开发考虑量子物理的控制理论（例如， $[9-41]$$[9-41]$[9-41][9-41] ）。

What types of quantum control can be envisaged? As for classical (i.e., nonquantum) systems, we distinguish between open- and closed-loop control.
可以设想哪些类型的量子控制？对于经典（即非量子）系统，我们区分开环控制和闭环控制。
Open-loop control has its usual meaning-a predetermined classical control signal is applied to the plant, in this case a quantum system, and no feedback is involved (Figure 31.3).
开环控制具有其通常的含义——一个预定的经典控制信号被施加到系统中，在这种情况下是量子系统，并且不涉及反馈（图 31.3）。

Closed-loop or feedback control also has its usual meaning-control actions depend on information gained while the plant is operating-however, care must be taken as to the nature of the controller
闭环或反馈控制也有其通常的含义——控制动作依赖于在工厂运行时获得的信息——然而，必须注意控制器的性质

FIGURE 31.2 Model of an atom.
图 31.2 原子的模型。

FIGURE 31.3 Open-loop control.
图 31.3 开环控制。

FIGURE 31.4 Closed-loop measurement feedback control.
图 31.4 闭环测量反馈控制。
and what is meant by “information.” When the controller is a classical system, which can only process classical information, some form of measurement of the quantum plant is needed, see Figure 31.4. This is called measurement feedback quantum control.
当控制器是一个经典系统，只能处理经典信息时，需要对量子植物进行某种形式的测量，见图 31.4。这被称为测量反馈量子控制。
The theory and applications that have been developed for measurement feedback depend on quantum filtering theory [42,43], as we explain in Sections 31.5.2 through 31.5.4. Measurement feedback quantum control can be effective for a wide range of applications, and has the benefit that the control algorithms can be implemented in conventional classical hardware (provided it is fast enough).
测量反馈的理论和应用依赖于量子滤波理论[42,43]，正如我们在 31.5.2 至 31.5.4 节中所解释的。测量反馈量子控制在广泛的应用中都可以有效，并且其优点在于控制算法可以在常规经典硬件中实现（前提是其速度足够快）。

It is also possible to use another quantum system as the controller, see Figure 31.5. This type of feedback does not use measurement, and the information flowing in the loop is fully quantum.
也可以使用另一个量子系统作为控制器，见图 31.5。这种反馈不使用测量，环路中流动的信息完全是量子信息。
This exchange of quantum information may be directional, via a quantum signal (such as a beam of light), or bidirectional, via a direct physical coupling. This is called coherent or quantum feedback quantum control.
这种量子信息的交换可能是单向的，通过量子信号（例如光束），或是双向的，通过直接的物理耦合。这被称为相干或量子反馈量子控制。
While quantum feedback is conceptually simple, at present little is known about how to systematically design fully ( CF ) coherent feedback loops. In Section 31.5.5, we describe one recent example of coherent feedback design.
尽管量子反馈在概念上很简单，但目前对如何系统地设计完全（CF）相干反馈回路知之甚少。在第 31.5.5 节中，我们描述了一个最近的相干反馈设计示例。
The benefits of coherent feedback include the preservation of quantum information, and that the timescales of the controller can be better matched to the quantum plant (which could have very fast dynamics).
一致反馈的好处包括量子信息的保存，以及控制器的时间尺度可以更好地与量子系统（可能具有非常快的动态）匹配。

FIGURE 31.5 Closed-loop feedback control with no measurement.
图 31.5 无测量的闭环反馈控制。

Let $\mathcal{H}$$\mathcal{H}$H\mathcal{H} be a separable Hilbert space with inner product $⟨\cdot ,\cdot ⟩$$⟨\cdot ,\cdot ⟩$(:*,*:)\langle\cdot, \cdot\rangle (taken to be linear in the second argument and conjugate linear in the first) and norm $\Vert \psi \Vert =\sqrt{⟨\psi ,\psi ⟩}$$\Vert \psi \Vert =\sqrt{⟨\psi ,\psi ⟩}$||psi||=sqrt((:psi,psi:))\|\psi\|=\sqrt{\langle\psi, \psi\rangle}. Basic examples are (i) $\mathcal{H}={\mathbb{C}}^n$$\mathcal{H}={\mathbb{C}}^n$H=C^(n)\mathcal{H}=\mathbb{C}^{n}, the space of $n$$n$nn-dimensional complex vectors, where $⟨\psi ,\varphi ⟩=\sum _{k=1}^n{\psi }_k^{†}{\varphi }_k$$⟨\psi ,\varphi ⟩=\sum _{k=1}^n {\psi }_k^{†}{\varphi }_k$(:psi,phi:)=sum_(k=1)^(n)psi_(k)^(†)phi_(k)\langle\psi, \phi\rangle=\sum_{k=1}^{n} \psi_{k}^{\dagger} \phi_{k}, where ${\psi }_k^{†}$${\psi }_k^{†}$psi_(k)^(†)\psi_{k}^{\dagger} denotes the adjoint (complex conjugate) of ${\psi }_k$${\psi }_k$psi_(k)\psi_{k}, and (ii) $\mathcal{H}=L^2(\mathbb{R})$$\mathcal{H}=L^2(\mathbb{R})$H=L^(2)(R)\mathcal{H}=L^{2}(\mathbb{R}), the space of complex-valued functions on $\mathbb{R}$$\mathbb{R}$R\mathbb{R} that have square integrable components, with inner product $⟨\psi ,\varphi ⟩=\int {\psi }^{†}(x)\varphi (x)dx$$⟨\psi ,\varphi ⟩=\int {\psi }^{†}(x)\varphi (x)dx$(:psi,phi:)=intpsi^(†)(x)phi(x)dx\langle\psi, \phi\rangle=\int \psi^{\dagger}(x) \phi(x) d x.
让 $\mathcal{H}$$\mathcal{H}$H\mathcal{H} 是一个可分的希尔伯特空间，内积为 $⟨\cdot ,\cdot ⟩$$⟨\cdot ,\cdot ⟩$(:*,*:)\langle\cdot, \cdot\rangle （在第二个参数上取线性，在第一个参数上取共轭线性），范数为 $\Vert \psi \Vert =\sqrt{⟨\psi ,\psi ⟩}$$\Vert \psi \Vert =\sqrt{⟨\psi ,\psi ⟩}$||psi||=sqrt((:psi,psi:))\|\psi\|=\sqrt{\langle\psi, \psi\rangle} 。基本例子有（i） $\mathcal{H}={\mathbb{C}}^n$$\mathcal{H}={\mathbb{C}}^n$H=C^(n)\mathcal{H}=\mathbb{C}^{n} ，即 $n$$n$nn 维复向量空间，其中 $⟨\psi ,\varphi ⟩=\sum _{k=1}^n{\psi }_k^{†}{\varphi }_k$$⟨\psi ,\varphi ⟩=\sum _{k=1}^n {\psi }_k^{†}{\varphi }_k$(:psi,phi:)=sum_(k=1)^(n)psi_(k)^(†)phi_(k)\langle\psi, \phi\rangle=\sum_{k=1}^{n} \psi_{k}^{\dagger} \phi_{k} ， ${\psi }_k^{†}$${\psi }_k^{†}$psi_(k)^(†)\psi_{k}^{\dagger} 表示 ${\psi }_k$${\psi }_k$psi_(k)\psi_{k} 的伴随（复共轭），以及（ii） $\mathcal{H}=L^2(\mathbb{R})$$\mathcal{H}=L^2(\mathbb{R})$H=L^(2)(R)\mathcal{H}=L^{2}(\mathbb{R}) ，即在 $\mathbb{R}$$\mathbb{R}$R\mathbb{R} 上具有平方可积分量的复值函数空间，内积为 $⟨\psi ,\varphi ⟩=\int {\psi }^{†}(x)\varphi (x)dx$$⟨\psi ,\varphi ⟩=\int {\psi }^{†}(x)\varphi (x)dx$(:psi,phi:)=intpsi^(†)(x)phi(x)dx\langle\psi, \phi\rangle=\int \psi^{\dagger}(x) \phi(x) d x 。

Let $\mathcal{B}(\mathcal{H})$$\mathcal{B}(\mathcal{H})$B(H)\mathscr{B}(\mathcal{H}) be the Banach space of bounded operators $A:\mathcal{H}\to \mathcal{H}$$A:\mathcal{H}\to \mathcal{H}$A:HrarrHA: \mathcal{H} \rightarrow \mathcal{H}. The commutator of two operators is defined by $[A,B]=AB-BA$$[A,B]=AB-BA$[A,B]=AB-BA[A, B]=A B-B A. For any $A\in \mathcal{B}(\mathcal{H})$$A\in \mathcal{B}(\mathcal{H})$A inB(H)A \in \mathscr{B}(\mathcal{H}), its adjoint $A^{†}\in \mathcal{B}(\mathcal{H})$$A^{†}\in \mathcal{B}(\mathcal{H})$A^(†)inB(H)A^{\dagger} \in \mathscr{B}(\mathcal{H}) is an operator defined by $⟨A^{†}\psi ,\varphi ⟩=⟨\psi ,A\varphi ⟩$$⟨A^{†}\psi ,\varphi ⟩=⟨\psi ,A\varphi ⟩$(:A^(†)psi,phi:)=(:psi,A phi:)\left\langle A^{\dagger} \psi, \phi\right\rangle=\langle\psi, A \phi\rangle for all $\psi ,\varphi \in \mathcal{H}$$\psi ,\varphi \in \mathcal{H}$psi,phi inH\psi, \phi \in \mathcal{H}. An operator $A\in \mathcal{B}(\mathcal{H})$$A\in \mathcal{B}(\mathcal{H})$A inB(H)A \in \mathscr{B}(\mathcal{H}) is called normal if $A{A}^{†}=A^{†}A$$A{A}^{†}=A^{†}A$AA^(†)=A^(†)AA A^{\dagger}=A^{\dagger} A. Two important types of normal operators are self-adjoint $(A=A^{†})$$\left(A=A^{†}\right)$(A=A^(†))\left(A=A^{\dagger}\right) and unitary $(A^{†}=A^{-1})$$\left(A^{†}=A^{-1}\right)$(A^(†)=A^(-1))\left(A^{\dagger}=A^{-1}\right). The spectral theorem for a normal operator $A$$A$AA says that (discrete case) there exists a complete set of orthonormal eigenvectors (such a set forms a basis for $\mathcal{H}$$\mathcal{H}$H\mathcal{H} ), and $A$$A$AA can be written as $A=\sum _n{a}_n{P}_n$$A=\sum _n a_n{P}_n$A=sum_(n)a_(n)P_(n)A=\sum_{n} a_{n} P_{n}, where $P_n$$P_n$P_(n)P_{n} is the projection onto the $n$$n$nnth eigenspace (diagonal representation) with associated eigenvalue $a_n$$a_n$a_(n)a_{n}. In Dirac’s bra-ket notation, the eigenvectors are written $|n⟩$$|n⟩$|n:)|n\rangle and the projections $P_n=|n⟩⟨n|$$P_n=|n⟩⟨n|$P_(n)=|n:)(:n|P_{n}=|n\rangle\langle n|. The projections resolve the identity (orthogonally): $\sum _n{P}_n=I$$\sum _n P_n=I$sum_(n)P_(n)=I\sum_{n} P_{n}=I. If $A$$A$AA is self-adjoint, the eigenvalues $a_n$$a_n$a_(n)a_{n} are all real. In this notation, most often written by physicists, the “ket” $|\psi ⟩$$|\psi ⟩$|psi:)|\psi\rangle is always a unit vector with adjoint $⟨\psi |$$⟨\psi |$(:psi|\langle\psi|. The norm of $|\psi ⟩$$|\psi ⟩$|psi:)|\psi\rangle is thus $\Vert \psi \Vert =\sqrt{⟨\psi \mid \psi ⟩}=1$$\Vert \psi \Vert =\sqrt{⟨\psi \mid \psi ⟩}=1$||psi||=sqrt((:psi∣psi:))=1\|\psi\|=\sqrt{\langle\psi \mid \psi\rangle}=1. (We will not always adhere to the ket-notation, and so sometimes write $\psi$$\psi$psi\psi and implicitly assume that it is a unit vector, i.e., ${\psi }^{†}\psi =1$${\psi }^{†}\psi =1$psi^(†)psi=1\psi^{\dagger} \psi=1.)
让 $\mathcal{B}(\mathcal{H})$$\mathcal{B}(\mathcal{H})$B(H)\mathscr{B}(\mathcal{H}) 成为有界算子的巴拿赫空间 $A:\mathcal{H}\to \mathcal{H}$$A:\mathcal{H}\to \mathcal{H}$A:HrarrHA: \mathcal{H} \rightarrow \mathcal{H} 。两个算子的对易子由 $[A,B]=AB-BA$$[A,B]=AB-BA$[A,B]=AB-BA[A, B]=A B-B A 定义。对于任何 $A\in \mathcal{B}(\mathcal{H})$$A\in \mathcal{B}(\mathcal{H})$A inB(H)A \in \mathscr{B}(\mathcal{H}) ，其伴随算子 $A^{†}\in \mathcal{B}(\mathcal{H})$$A^{†}\in \mathcal{B}(\mathcal{H})$A^(†)inB(H)A^{\dagger} \in \mathscr{B}(\mathcal{H}) 是由 $⟨A^{†}\psi ,\varphi ⟩=⟨\psi ,A\varphi ⟩$$⟨A^{†}\psi ,\varphi ⟩=⟨\psi ,A\varphi ⟩$(:A^(†)psi,phi:)=(:psi,A phi:)\left\langle A^{\dagger} \psi, \phi\right\rangle=\langle\psi, A \phi\rangle 定义的，适用于所有 $\psi ,\varphi \in \mathcal{H}$$\psi ,\varphi \in \mathcal{H}$psi,phi inH\psi, \phi \in \mathcal{H} 。如果算子 $A\in \mathcal{B}(\mathcal{H})$$A\in \mathcal{B}(\mathcal{H})$A inB(H)A \in \mathscr{B}(\mathcal{H}) 满足 $A{A}^{†}=A^{†}A$$A{A}^{†}=A^{†}A$AA^(†)=A^(†)AA A^{\dagger}=A^{\dagger} A ，则称其为正常算子。两种重要的正常算子是自伴随算子 $(A=A^{†})$$\left(A=A^{†}\right)$(A=A^(†))\left(A=A^{\dagger}\right) 和单位算子 $(A^{†}=A^{-1})$$\left(A^{†}=A^{-1}\right)$(A^(†)=A^(-1))\left(A^{\dagger}=A^{-1}\right) 。正常算子 $A$$A$AA 的谱定理表明（离散情况）存在一组完整的正交归一特征向量（这样的集合构成 $\mathcal{H}$$\mathcal{H}$H\mathcal{H} 的基），并且 $A$$A$AA 可以写成 $A=\sum _n{a}_n{P}_n$$A=\sum _n a_n{P}_n$A=sum_(n)a_(n)P_(n)A=\sum_{n} a_{n} P_{n} ，其中 $P_n$$P_n$P_(n)P_{n} 是投影到第 $n$$n$nn 个特征空间（对角表示），其相关特征值为 $a_n$$a_n$a_(n)a_{n} 。在迪拉克的布拉-凯特符号中，特征向量写作 $|n⟩$$|n⟩$|n:)|n\rangle ，投影写作 $P_n=|n⟩⟨n|$$P_n=|n⟩⟨n|$P_(n)=|n:)(:n|P_{n}=|n\rangle\langle n| 。这些投影正交地分解单位元： $\sum _n{P}_n=I$$\sum _n P_n=I$sum_(n)P_(n)=I\sum_{n} P_{n}=I 。如果 $A$$A$AA 是自伴随的，则特征值 $a_n$$a_n$a_(n)a_{n} 全部为实数。在这种符号中，物理学家通常写作的“凯特” $|\psi ⟩$$|\psi ⟩$|psi:)|\psi\rangle 始终是伴随 $⟨\psi |$$⟨\psi |$(:psi|\langle\psi| 的单位向量。因此， $|\psi ⟩$$|\psi ⟩$|psi:)|\psi\rangle 的范数为 $\Vert \psi \Vert =\sqrt{⟨\psi \mid \psi ⟩}=1$$\Vert \psi \Vert =\sqrt{⟨\psi \mid \psi ⟩}=1$||psi||=sqrt((:psi∣psi:))=1\|\psi\|=\sqrt{\langle\psi \mid \psi\rangle}=1 。（我们并不总是遵循凯特符号，有时写作 $\psi$$\psi$psi\psi 并隐含假设它是单位向量，即 ${\psi }^{†}\psi =1$${\psi }^{†}\psi =1$psi^(†)psi=1\psi^{\dagger} \psi=1 。）

Tensor products are used to describe composite systems. If ${\mathcal{H}}_1$${\mathcal{H}}_1$H_(1)\mathcal{H}_{1} and ${\mathcal{H}}_2$${\mathcal{H}}_2$H_(2)\mathcal{H}_{2} are Hilbert spaces, the tensor product ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$${\mathcal{H}}_1\otimes {\mathcal{H}}_2$H_(1)oxH_(2)\mathcal{H}_{1} \otimes \mathcal{H}_{2} is the Hilbert space consisting of linear combinations of the form ${\psi }_1\otimes {\psi }_2$${\psi }_1\otimes {\psi }_2$psi_(1)oxpsi_(2)\psi_{1} \otimes \psi_{2}, and inner product $⟨{\psi }_1\otimes {\psi }_2,{\varphi }_1\otimes {\varphi }_2⟩=⟨{\psi }_1,{\varphi }_1⟩⟨{\psi }_2,{\varphi }_2⟩$$⟨{\psi }_1\otimes {\psi }_2,{\varphi }_1\otimes {\varphi }_2⟩=⟨{\psi }_1,{\varphi }_1⟩⟨{\psi }_2,{\varphi }_2⟩$(:psi_(1)oxpsi_(2),phi_(1)oxphi_(2):)=(:psi_(1),phi_(1):)(:psi_(2),phi_(2):)\left\langle\psi_{1} \otimes \psi_{2}, \phi_{1} \otimes \phi_{2}\right\rangle=\left\langle\psi_{1}, \phi_{1}\right\rangle\left\langle\psi_{2}, \phi_{2}\right\rangle. Here, ${\psi }_1,{\varphi }_1\in {\mathcal{H}}_1$${\psi }_1,{\varphi }_1\in {\mathcal{H}}_1$psi_(1),phi_(1)inH_(1)\psi_{1}, \phi_{1} \in \mathcal{H}_{1} and ${\psi }_2,{\varphi }_2\in {\mathcal{H}}_2$${\psi }_2,{\varphi }_2\in {\mathcal{H}}_2$psi_(2),phi_(2)inH_(2)\psi_{2}, \phi_{2} \in \mathcal{H}_{2}. If $A_1$$A_1$A_(1)A_{1} and $A_2$$A_2$A_(2)A_{2} are operators on ${\mathcal{H}}_1$${\mathcal{H}}_1$H_(1)\mathcal{H}_{1} and ${\mathcal{H}}_2$${\mathcal{H}}_2$H_(2)\mathcal{H}_{2}, respectively, then $A_1\otimes A_2$$A_1\otimes A_2$A_(1)oxA_(2)A_{1} \otimes A_{2} is an operator on ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$${\mathcal{H}}_1\otimes {\mathcal{H}}_2$H_(1)oxH_(2)\mathcal{H}_{1} \otimes \mathcal{H}_{2} and is defined by $(A_1\otimes A_2)({\psi }_1\otimes {\psi }_2)=A_1{\psi }_1\otimes A_2{\psi }_2$$\left(A_1\otimes A_2\right)\left({\psi }_1\otimes {\psi }_2\right)=A_1{\psi }_1\otimes A_2{\psi }_2$(A_(1)oxA_(2))(psi_(1)oxpsi_(2))=A_(1)psi_(1)oxA_(2)psi_(2)\left(A_{1} \otimes A_{2}\right)\left(\psi_{1} \otimes \psi_{2}\right)=A_{1} \psi_{1} \otimes A_{2} \psi_{2}. Often, $A_1\otimes A_2$$A_1\otimes A_2$A_(1)oxA_(2)A_{1} \otimes A_{2} is written $A_1{A}_2$$A_1{A}_2$A_(1)A_(2)A_{1} A_{2}.
张量积用于描述复合系统。如果 ${\mathcal{H}}_1$${\mathcal{H}}_1$H_(1)\mathcal{H}_{1} 和 ${\mathcal{H}}_2$${\mathcal{H}}_2$H_(2)\mathcal{H}_{2} 是希尔伯特空间，则张量积 ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$${\mathcal{H}}_1\otimes {\mathcal{H}}_2$H_(1)oxH_(2)\mathcal{H}_{1} \otimes \mathcal{H}_{2} 是由形式为 ${\psi }_1\otimes {\psi }_2$${\psi }_1\otimes {\psi }_2$psi_(1)oxpsi_(2)\psi_{1} \otimes \psi_{2} 的线性组合构成的希尔伯特空间，内积为 $⟨{\psi }_1\otimes {\psi }_2,{\varphi }_1\otimes {\varphi }_2⟩=⟨{\psi }_1,{\varphi }_1⟩⟨{\psi }_2,{\varphi }_2⟩$$⟨{\psi }_1\otimes {\psi }_2,{\varphi }_1\otimes {\varphi }_2⟩=⟨{\psi }_1,{\varphi }_1⟩⟨{\psi }_2,{\varphi }_2⟩$(:psi_(1)oxpsi_(2),phi_(1)oxphi_(2):)=(:psi_(1),phi_(1):)(:psi_(2),phi_(2):)\left\langle\psi_{1} \otimes \psi_{2}, \phi_{1} \otimes \phi_{2}\right\rangle=\left\langle\psi_{1}, \phi_{1}\right\rangle\left\langle\psi_{2}, \phi_{2}\right\rangle 。这里， ${\psi }_1,{\varphi }_1\in {\mathcal{H}}_1$${\psi }_1,{\varphi }_1\in {\mathcal{H}}_1$psi_(1),phi_(1)inH_(1)\psi_{1}, \phi_{1} \in \mathcal{H}_{1} 和 ${\psi }_2,{\varphi }_2\in {\mathcal{H}}_2$${\psi }_2,{\varphi }_2\in {\mathcal{H}}_2$psi_(2),phi_(2)inH_(2)\psi_{2}, \phi_{2} \in \mathcal{H}_{2} 。如果 $A_1$$A_1$A_(1)A_{1} 和 $A_2$$A_2$A_(2)A_{2} 分别是作用于 ${\mathcal{H}}_1$${\mathcal{H}}_1$H_(1)\mathcal{H}_{1} 和 ${\mathcal{H}}_2$${\mathcal{H}}_2$H_(2)\mathcal{H}_{2} 的算子，则 $A_1\otimes A_2$$A_1\otimes A_2$A_(1)oxA_(2)A_{1} \otimes A_{2} 是作用于 ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$${\mathcal{H}}_1\otimes {\mathcal{H}}_2$H_(1)oxH_(2)\mathcal{H}_{1} \otimes \mathcal{H}_{2} 的算子，定义为 $(A_1\otimes A_2)({\psi }_1\otimes {\psi }_2)=A_1{\psi }_1\otimes A_2{\psi }_2$$\left(A_1\otimes A_2\right)\left({\psi }_1\otimes {\psi }_2\right)=A_1{\psi }_1\otimes A_2{\psi }_2$(A_(1)oxA_(2))(psi_(1)oxpsi_(2))=A_(1)psi_(1)oxA_(2)psi_(2)\left(A_{1} \otimes A_{2}\right)\left(\psi_{1} \otimes \psi_{2}\right)=A_{1} \psi_{1} \otimes A_{2} \psi_{2} 。通常， $A_1\otimes A_2$$A_1\otimes A_2$A_(1)oxA_(2)A_{1} \otimes A_{2} 写作 $A_1{A}_2$$A_1{A}_2$A_(1)A_(2)A_{1} A_{2} 。

In quantum mechanics [44] physical quantities like energy, spin, position, and so on, are expressed as observables; these are represented as self-adjoint operators $(A=A^{†})$$\left(A=A^{†}\right)$(A=A^(†))\left(A=A^{\dagger}\right) acting on a Hilbert space $\mathcal{H}$$\mathcal{H}$H\mathcal{H}.
在量子力学中，能量、自旋、位置等物理量被表示为可观测量；这些量被表示为作用在希尔伯特空间上的自伴算符。

The state of a quantum system is a unit vector $\psi \in \mathcal{H}$$\psi \in \mathcal{H}$psi inH\psi \in \mathcal{H} or $|\psi ⟩\in \mathcal{H}$$|\psi ⟩\in \mathcal{H}$|psi:)inH|\psi\rangle \in \mathcal{H}. In the discrete case every element ${\psi }_k$${\psi }_k$psi_(k)\psi_{k} is a possible state of the system with a probability of occurence ${\left|{\psi }_k\right|}^2$${\left|{\psi }_k\right|}^2$|psi_(k)|^(2)\left|\psi_{k}\right|^{2}. Hence, $\Vert \psi \Vert =1$$\Vert \psi \Vert =1$||psi||=1\|\psi\|=1 means that all outcomes can occur. (The same applies in the continuous case, for example, at a spatial point $r=(x,y,z)$$r=(x,y,z)$r=(x,y,z)r=(x, y, z), $\Vert \psi (r){\Vert }^{2}dxdydz$$\Vert \psi (r){\Vert }^{2}dxdydz$||psi(r)||^(2)dxdydz\|\psi(r)\|^{2} d x d y d z is the probability that a particle would be found in the differential volume.) The state $|\psi ⟩$$|\psi ⟩$|psi:)|\psi\rangle is called a pure state. Pure states are special cases of a more general notion of state referred to as a density operator or density matrix. A density operator $\rho$$\rho$rho\rho is a positive self-adjoint operator on $\mathcal{H}$$\mathcal{H}$H\mathcal{H} with trace one. Pure states are of the form $\rho =|\psi ⟩⟨\psi |$$\rho =|\psi ⟩⟨\psi |$rho=|psi:)(:psi|\rho=|\psi\rangle\langle\psi|. More generally, states that are convex combinations of pure states are called mixed states: $\rho =\sum _n{\lambda }_n|{\psi }_n⟩⟨{\psi }_n|$$\rho =\sum _n {\lambda }_n\left|{\psi }_n\right.⟩\left.⟨{\psi }_n\right|$rho=sum_(n)lambda_(n)|psi_(n):)(:psi_(n)|\rho=\sum_{n} \lambda_{n}\left|\psi_{n}\right\rangle\left\langle\psi_{n}\right|.
量子系统的状态是单位向量 $\psi \in \mathcal{H}$$\psi \in \mathcal{H}$psi inH\psi \in \mathcal{H} 或 $|\psi ⟩\in \mathcal{H}$$|\psi ⟩\in \mathcal{H}$|psi:)inH|\psi\rangle \in \mathcal{H} 。在离散情况下，每个元素 ${\psi }_k$${\psi }_k$psi_(k)\psi_{k} 都是系统的一个可能状态，发生的概率为 ${\left|{\psi }_k\right|}^2$${\left|{\psi }_k\right|}^2$|psi_(k)|^(2)\left|\psi_{k}\right|^{2} 。因此， $\Vert \psi \Vert =1$$\Vert \psi \Vert =1$||psi||=1\|\psi\|=1 意味着所有结果都可以发生。（在连续情况下也是如此，例如，在空间点 $r=(x,y,z)$$r=(x,y,z)$r=(x,y,z)r=(x, y, z) ， $\Vert \psi (r){\Vert }^{2}dxdydz$$\Vert \psi (r){\Vert }^{2}dxdydz$||psi(r)||^(2)dxdydz\|\psi(r)\|^{2} d x d y d z 是粒子在微分体积中被发现的概率。）状态 $|\psi ⟩$$|\psi ⟩$|psi:)|\psi\rangle 称为纯态。纯态是更一般的状态概念的特例，称为密度算子或密度矩阵。密度算子 $\rho$$\rho$rho\rho 是在 $\mathcal{H}$$\mathcal{H}$H\mathcal{H} 上的正自伴算子，迹为一。纯态的形式为 $\rho =|\psi ⟩⟨\psi |$$\rho =|\psi ⟩⟨\psi |$rho=|psi:)(:psi|\rho=|\psi\rangle\langle\psi| 。更一般地，纯态的凸组合称为混合态： $\rho =\sum _n{\lambda }_n|{\psi }_n⟩⟨{\psi }_n|$$\rho =\sum _n {\lambda }_n\left|{\psi }_n\right.⟩\left.⟨{\psi }_n\right|$rho=sum_(n)lambda_(n)|psi_(n):)(:psi_(n)|\rho=\sum_{n} \lambda_{n}\left|\psi_{n}\right\rangle\left\langle\psi_{n}\right| 。

The postulates of quantum mechanics state that for a closed system the evolution of states obeys the Schrödinger equation
量子力学的公设指出，对于一个封闭系统，状态的演化遵循薛定谔方程

Here, $H$$H$HH is an observable called the Hamiltonian, and represents the energy of the system, and $i=\sqrt{-1}$$i=\sqrt{-1}$i=sqrt(-1)i=\sqrt{-1}. Since $|\psi (t)⟩$$|\psi (t)⟩$|psi(t):)|\psi(t)\rangle has unit norm, the evolution from one time to another must be unitary, that is, $\psi (t)=$$\psi (t)=$psi(t)=\psi(t)= $U(t)\psi (0)$$U(t)\psi (0)$U(t)psi(0)U(t) \psi(0) where the unitary transition matrix, referred to as the propagator, obeys a matrix version of the Schrödinger equation, $i\dot{U}=HU,U(0)=I$$i\dot{U}=HU,U(0)=I$iU^(˙)=HU,U(0)=Ii \dot{U}=H U, U(0)=I (the identity). Density operators also evolve unitarily, $\rho (t)=U(t)\rho U^{†}(t)$$\rho (t)=U(t)\rho U^{†}(t)$rho(t)=U(t)rhoU^(†)(t)\rho(t)=U(t) \rho U^{\dagger}(t), so, by the Schrödinger equation (31.1) we have $i\dot{\rho }=[H,\rho ]=H\rho -\rho H$$i\dot{\rho }=[H,\rho ]=H\rho -\rho H$irho^(˙)=[H,rho]=H rho-rho Hi \dot{\rho}=[H, \rho]=H \rho-\rho H. We may view state vectors as fixed in time, while observables are taken to evolve according to $A(t)=U^{†}(t)AU(t)$$A(t)=U^{†}(t)AU(t)$A(t)=U^(†)(t)AU(t)A(t)=U^{\dagger}(t) A U(t) : this is the Heisenberg picture.
在这里， $H$$H$HH 是一个称为哈密顿量的可观测量，代表系统的能量，以及 $i=\sqrt{-1}$$i=\sqrt{-1}$i=sqrt(-1)i=\sqrt{-1} 。由于 $|\psi (t)⟩$$|\psi (t)⟩$|psi(t):)|\psi(t)\rangle 具有单位范数，从一个时间到另一个时间的演化必须是单位的，即 $\psi (t)=$$\psi (t)=$psi(t)=\psi(t)= $U(t)\psi (0)$$U(t)\psi (0)$U(t)psi(0)U(t) \psi(0) ，其中称为传播算子的单位转移矩阵遵循薛定谔方程的矩阵版本， $i\dot{U}=HU,U(0)=I$$i\dot{U}=HU,U(0)=I$iU^(˙)=HU,U(0)=Ii \dot{U}=H U, U(0)=I （单位矩阵）。密度算子也以单位方式演化， $\rho (t)=U(t)\rho U^{†}(t)$$\rho (t)=U(t)\rho U^{†}(t)$rho(t)=U(t)rhoU^(†)(t)\rho(t)=U(t) \rho U^{\dagger}(t) ，因此，根据薛定谔方程（31.1），我们有 $i\dot{\rho }=[H,\rho ]=H\rho -\rho H$$i\dot{\rho }=[H,\rho ]=H\rho -\rho H$irho^(˙)=[H,rho]=H rho-rho Hi \dot{\rho}=[H, \rho]=H \rho-\rho H 。我们可以将状态向量视为固定在时间上，而可观测量则根据 $A(t)=U^{†}(t)AU(t)$$A(t)=U^{†}(t)AU(t)$A(t)=U^(†)(t)AU(t)A(t)=U^{\dagger}(t) A U(t) 演化：这就是海森堡图像。

The numerical value of a measurement of $A$$A$AA is an eigenvalue of $A$$A$AA. If the system is in state $\rho$$\rho$rho\rho at the time of the measurement, and $A$$A$AA has the spectral decomposition $A=\sum _n{a}_n{P}_n$$A=\sum _n a_n{P}_n$A=sum_(n)a_(n)P_(n)A=\sum_{n} a_{n} P_{n}, the value $a_n$$a_n$a_(n)a_{n} occurs with probability
测量 $A$$A$AA 的数值是 $A$$A$AA 的特征值。如果系统在测量时处于状态 $\rho$$\rho$rho\rho ，并且 $A$$A$AA 具有谱分解 $A=\sum _n{a}_n{P}_n$$A=\sum _n a_n{P}_n$A=sum_(n)a_(n)P_(n)A=\sum_{n} a_{n} P_{n} ，则值 $a_n$$a_n$a_(n)a_{n} 以概率出现。