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] ). 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] )。
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。
31.2 Some Quantum Mechanics 31.2 一些量子力学
31.2.1 Preliminaries 31.2.1 初步准备
Let H\mathcal{H} be a separable Hilbert space with inner product (:*,*:)\langle\cdot, \cdot\rangle (taken to be linear in the second argument and conjugate linear in the first) and norm ||psi||=sqrt((:psi,psi:))\|\psi\|=\sqrt{\langle\psi, \psi\rangle}. Basic examples are (i) H=C^(n)\mathcal{H}=\mathbb{C}^{n}, the space of nn-dimensional complex vectors, where (: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}^{\dagger} denotes the adjoint (complex conjugate) of psi_(k)\psi_{k}, and (ii) H=L^(2)(R)\mathcal{H}=L^{2}(\mathbb{R}), the space of complex-valued functions on R\mathbb{R} that have square integrable components, with inner product (:psi,phi:)=intpsi^(†)(x)phi(x)dx\langle\psi, \phi\rangle=\int \psi^{\dagger}(x) \phi(x) d x. 让 H\mathcal{H} 是一个可分的希尔伯特空间,内积为 (:*,*:)\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 ||psi||=sqrt((:psi,psi:))\|\psi\|=\sqrt{\langle\psi, \psi\rangle} 。基本例子有(i) H=C^(n)\mathcal{H}=\mathbb{C}^{n} ,即 nn 维复向量空间,其中 (: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}^{\dagger} 表示 psi_(k)\psi_{k} 的伴随(复共轭),以及(ii) H=L^(2)(R)\mathcal{H}=L^{2}(\mathbb{R}) ,即在 R\mathbb{R} 上具有平方可积分量的复值函数空间,内积为 (:psi,phi:)=intpsi^(†)(x)phi(x)dx\langle\psi, \phi\rangle=\int \psi^{\dagger}(x) \phi(x) d x 。
Let B(H)\mathscr{B}(\mathcal{H}) be the Banach space of bounded operators A:HrarrHA: \mathcal{H} \rightarrow \mathcal{H}. The commutator of two operators is defined by [A,B]=AB-BA[A, B]=A B-B A. For any A inB(H)A \in \mathscr{B}(\mathcal{H}), its adjoint A^(†)inB(H)A^{\dagger} \in \mathscr{B}(\mathcal{H}) is an operator defined by (:A^(†)psi,phi:)=(:psi,A phi:)\left\langle A^{\dagger} \psi, \phi\right\rangle=\langle\psi, A \phi\rangle for all psi,phi inH\psi, \phi \in \mathcal{H}. An operator A inB(H)A \in \mathscr{B}(\mathcal{H}) is called normal if AA^(†)=A^(†)AA A^{\dagger}=A^{\dagger} A. Two important types of normal operators are self-adjoint (A=A^(†))\left(A=A^{\dagger}\right) and unitary (A^(†)=A^(-1))\left(A^{\dagger}=A^{-1}\right). The spectral theorem for a normal operator AA says that (discrete case) there exists a complete set of orthonormal eigenvectors (such a set forms a basis for H\mathcal{H} ), and AA can be written as A=sum_(n)a_(n)P_(n)A=\sum_{n} a_{n} P_{n}, where P_(n)P_{n} is the projection onto the nnth eigenspace (diagonal representation) with associated eigenvalue a_(n)a_{n}. In Dirac’s bra-ket notation, the eigenvectors are written |n:)|n\rangle and the projections 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. If AA is self-adjoint, the eigenvalues a_(n)a_{n} are all real. In this notation, most often written by physicists, the “ket” |psi:)|\psi\rangle is always a unit vector with adjoint (:psi|\langle\psi|. The norm of |psi:)|\psi\rangle is thus ||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 and implicitly assume that it is a unit vector, i.e., psi^(†)psi=1\psi^{\dagger} \psi=1.) 让 B(H)\mathscr{B}(\mathcal{H}) 成为有界算子的巴拿赫空间 A:HrarrHA: \mathcal{H} \rightarrow \mathcal{H} 。两个算子的对易子由 [A,B]=AB-BA[A, B]=A B-B A 定义。对于任何 A inB(H)A \in \mathscr{B}(\mathcal{H}) ,其伴随算子 A^(†)inB(H)A^{\dagger} \in \mathscr{B}(\mathcal{H}) 是由 (: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,phi inH\psi, \phi \in \mathcal{H} 。如果算子 A inB(H)A \in \mathscr{B}(\mathcal{H}) 满足 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^{\dagger}\right) 和单位算子 (A^(†)=A^(-1))\left(A^{\dagger}=A^{-1}\right) 。正常算子 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 H\mathcal{H} 的基),并且 AA 可以写成 A=sum_(n)a_(n)P_(n)A=\sum_{n} a_{n} P_{n} ,其中 P_(n)P_{n} 是投影到第 nn 个特征空间(对角表示),其相关特征值为 a_(n)a_{n} 。在迪拉克的布拉-凯特符号中,特征向量写作 |n:)|n\rangle ,投影写作 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 。如果 AA 是自伴随的,则特征值 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\rangle 始终是伴随 (:psi|\langle\psi| 的单位向量。因此, |psi:)|\psi\rangle 的范数为 ||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 并隐含假设它是单位向量,即 psi^(†)psi=1\psi^{\dagger} \psi=1 。)
Tensor products are used to describe composite systems. If H_(1)\mathcal{H}_{1} and H_(2)\mathcal{H}_{2} are Hilbert spaces, the tensor product H_(1)oxH_(2)\mathcal{H}_{1} \otimes \mathcal{H}_{2} is the Hilbert space consisting of linear combinations of the form psi_(1)oxpsi_(2)\psi_{1} \otimes \psi_{2}, and inner product (: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),phi_(1)inH_(1)\psi_{1}, \phi_{1} \in \mathcal{H}_{1} and psi_(2),phi_(2)inH_(2)\psi_{2}, \phi_{2} \in \mathcal{H}_{2}. If A_(1)A_{1} and A_(2)A_{2} are operators on H_(1)\mathcal{H}_{1} and H_(2)\mathcal{H}_{2}, respectively, then A_(1)oxA_(2)A_{1} \otimes A_{2} is an operator on H_(1)oxH_(2)\mathcal{H}_{1} \otimes \mathcal{H}_{2} and is defined by (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)oxA_(2)A_{1} \otimes A_{2} is written A_(1)A_(2)A_{1} A_{2}. 张量积用于描述复合系统。如果 H_(1)\mathcal{H}_{1} 和 H_(2)\mathcal{H}_{2} 是希尔伯特空间,则张量积 H_(1)oxH_(2)\mathcal{H}_{1} \otimes \mathcal{H}_{2} 是由形式为 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)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),phi_(1)inH_(1)\psi_{1}, \phi_{1} \in \mathcal{H}_{1} 和 psi_(2),phi_(2)inH_(2)\psi_{2}, \phi_{2} \in \mathcal{H}_{2} 。如果 A_(1)A_{1} 和 A_(2)A_{2} 分别是作用于 H_(1)\mathcal{H}_{1} 和 H_(2)\mathcal{H}_{2} 的算子,则 A_(1)oxA_(2)A_{1} \otimes A_{2} 是作用于 H_(1)oxH_(2)\mathcal{H}_{1} \otimes \mathcal{H}_{2} 的算子,定义为 (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)oxA_(2)A_{1} \otimes A_{2} 写作 A_(1)A_(2)A_{1} A_{2} 。
31.2.2 The Postulates of Quantum Mechanics 31.2.2 量子力学的公设
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^{\dagger}\right) acting on a Hilbert space 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 inH\psi \in \mathcal{H} or |psi:)inH|\psi\rangle \in \mathcal{H}. In the discrete case every element psi_(k)\psi_{k} is a possible state of the system with a probability of occurence |psi_(k)|^(2)\left|\psi_{k}\right|^{2}. Hence, ||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), ||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\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 is a positive self-adjoint operator on H\mathcal{H} with trace one. Pure states are of the form 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\rangle\left\langle\psi_{n}\right|. 量子系统的状态是单位向量 psi inH\psi \in \mathcal{H} 或 |psi:)inH|\psi\rangle \in \mathcal{H} 。在离散情况下,每个元素 psi_(k)\psi_{k} 都是系统的一个可能状态,发生的概率为 |psi_(k)|^(2)\left|\psi_{k}\right|^{2} 。因此, ||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) , ||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\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 是在 H\mathcal{H} 上的正自伴算子,迹为一。纯态的形式为 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\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, HH is an observable called the Hamiltonian, and represents the energy of the system, and i=sqrt(-1)i=\sqrt{-1}. Since |psi(t):)|\psi(t)\rangle has unit norm, the evolution from one time to another must be unitary, that is, psi(t)=\psi(t)=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, iU^(˙)=HU,U(0)=Ii \dot{U}=H U, U(0)=I (the identity). Density operators also evolve unitarily, rho(t)=U(t)rhoU^(†)(t)\rho(t)=U(t) \rho U^{\dagger}(t), so, by the Schrödinger equation (31.1) we have 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^{\dagger}(t) A U(t) : this is the Heisenberg picture. 在这里, 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} 。由于 |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)=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, 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)rhoU^(†)(t)\rho(t)=U(t) \rho U^{\dagger}(t) ,因此,根据薛定谔方程(31.1),我们有 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^{\dagger}(t) A U(t) 演化:这就是海森堡图像。
The numerical value of a measurement of AA is an eigenvalue of AA. If the system is in state rho\rho at the time of the measurement, and AA has the spectral decomposition A=sum_(n)a_(n)P_(n)A=\sum_{n} a_{n} P_{n}, the value a_(n)a_{n} occurs with probability 测量 AA 的数值是 AA 的特征值。如果系统在测量时处于状态 rho\rho ,并且 AA 具有谱分解 A=sum_(n)a_(n)P_(n)A=\sum_{n} a_{n} P_{n} ,则值 a_(n)a_{n} 以概率出现。
This is Von Neumann’s state reduction. When a quantum system is in a pure state |psi:)|\psi\rangle, the expected value of an observable AA is defined in terms of the Hilbert space inner product: (:A:)=(:psi,A psi:)\langle A\rangle=\langle\psi, A \psi\rangle. Using the spectral decomposition of AA gives (:A:)=sum_(n)lambda_(n)(:psi,P_(n)psi:)\langle A\rangle=\sum_{n} \lambda_{n}\left\langle\psi, P_{n} \psi\right\rangle. If the system is in the mixed state rho\rho, then (:A:)=Tr(A rho)=sum_(n)lambda_(n)Tr(P_(n)rho)\langle A\rangle=\operatorname{Tr}(A \rho)=\sum_{n} \lambda_{n} \operatorname{Tr}\left(P_{n} \rho\right). (We will use the notations (:A:)\langle A\rangle or P[A]\mathbb{P}[A] for the expected value of an observable A.) 这是冯·诺依曼的状态简化。当量子系统处于纯态 |psi:)|\psi\rangle 时,可观测量 AA 的期望值通过希尔伯特空间内积定义为 (:A:)=(:psi,A psi:)\langle A\rangle=\langle\psi, A \psi\rangle 。使用 AA 的谱分解得到 (:A:)=sum_(n)lambda_(n)(:psi,P_(n)psi:)\langle A\rangle=\sum_{n} \lambda_{n}\left\langle\psi, P_{n} \psi\right\rangle 。如果系统处于混合态 rho\rho ,则 (:A:)=Tr(A rho)=sum_(n)lambda_(n)Tr(P_(n)rho)\langle A\rangle=\operatorname{Tr}(A \rho)=\sum_{n} \lambda_{n} \operatorname{Tr}\left(P_{n} \rho\right) 。(我们将使用符号 (:A:)\langle A\rangle 或 P[A]\mathbb{P}[A] 表示可观测量 A 的期望值。)
In a more abstract mathematical sense, if C\mathscr{C} is a commutative collection of operators (a commutative *-algebra), then by the spectral theorem [13, Theorem 2.4] a density operator rho\rho determines a classical probability distribution P\mathbf{P} and for all C inCC \in \mathscr{C} a classical random variable iota(C)\iota(C) on a classical probability space constructed from the spectrum of C\mathscr{C} such that P[C]=P[iota(C)]\mathbb{P}[C]=\mathbf{P}[\iota(C)]. When the context is clear we may abuse the notation and simply write CC for both the observable C inCC \in \mathscr{C} or the corresponding classical random variable iota(C)\iota(C). In case of the postulate stated above for an observable AA, the projections P_(n)P_{n} generate such a commutative collection C\mathscr{C}. 在更抽象的数学意义上,如果 C\mathscr{C} 是一个可交换的算子集合(一个可交换的*-代数),那么根据谱定理[13,定理 2.4],一个密度算子 rho\rho 确定了一个经典概率分布 P\mathbf{P} ,并且对于所有 C inCC \in \mathscr{C} ,在由 C\mathscr{C} 的谱构造的经典概率空间上有一个经典随机变量 iota(C)\iota(C) ,使得 P[C]=P[iota(C)]\mathbb{P}[C]=\mathbf{P}[\iota(C)] 。当上下文清晰时,我们可以滥用符号,简单地用 CC 表示可观测量 C inCC \in \mathscr{C} 或相应的经典随机变量 iota(C)\iota(C) 。在上述可观测量 AA 的公设情况下,投影 P_(n)P_{n} 生成这样一个可交换集合 C\mathscr{C} 。
31.2.3 Open Quantum Systems 31.2.3 开放量子系统
Open quantum systems are quantum systems that form part of a larger closed system. Figure 31.6 illustrates a representation of a system SS which is “open” to the environment EE. The environment consists 开放量子系统是构成更大封闭系统一部分的量子系统。图 31.6 展示了一个“开放”于环境的系统 SS 的表示。环境由以下组成:
FIGURE 31.6 Representation of an open quantum system. 图 31.6 开放量子系统的表示。
of an inaccessible part of the whole system, for example, a heat bath, nuclear spins, phonons, and so on. The complete SES E system (a composite system) obeys the normal evolutionary dynamics of quantum mechanics as given by a unitary U_(SE)U_{S E}, which may depend on externally applied classical controls. 一个整体系统中不可接触的部分,例如热浴、核自旋、声子等。完整的 SES E 系统(复合系统)遵循量子力学的正常演化动态,由一个单位 U_(SE)U_{S E} 给出,该单位可能依赖于外部施加的经典控制。
Here the state of the SS-system is accessible, while the state of the EE-system is not accessible. We will refer to the SS-system as the system and the the EE-system as the environment or bath. Since not every state of the universe is accessible, it is of basic interest to describe the potentially nonunitary transformation of the system state from one time to another. 在这里, SS -系统的状态是可访问的,而 EE -系统的状态是不可访问的。我们将 SS -系统称为系统,将 EE -系统称为环境或浴。由于宇宙的每个状态并非都是可访问的,因此描述系统状态从一个时间到另一个时间的潜在非单位变换是基本的兴趣所在。
开放系统的输出状态因此是单位动力学 U_(SE)U_{S E} 与 EE -系统对 SS -系统的平均影响的结合[45]。
然而,在其他情况下, SS -系统的状态是不可获取的,而 EE -系统或 EE -系统的部分是可获取的,并可用于反馈控制。例如,在图 31.7 中, SS -系统是一个原子,而 EE -系统是外部自由移动的场。原子无法直接测量。相反,测量场的可观察量 y_(0)(t)y_{0}(t) ,并可以通过经典方式处理,并用于测量反馈控制,见第 31.5.3 节。或者,场不必被测量,而可以通过另一个量子系统进行相干处理,如在相干反馈控制中,见第 31.5.5 节。
31.2.4 凸性与量子力学
凸性在量子力学中非常自然地出现,并在量子估计中发挥着重要作用,许多问题可以被表述为凸优化。
考虑例如以下凸集,这些凸集源自量子力学在维度为 nn 的希尔伯特空间中的一些基本方面:
图 31.7 腔体中的原子反馈控制。在第 31.2.3 节的符号中,原子是 SS -系统,而场 b_("in ")(t)b_{\text {in }}(t) 和 b_("out ")(t)b_{\text {out }}(t) 构成了 EE -系统。在这里, EE -系统的一个可观测量 y_(0)(t)y_{0}(t) 被测量并用于反馈回路。
Moreover, the set {O_(alpha gamma)}\left\{O_{\alpha \gamma}\right\} is a POVM. If Q_(gamma)Q_{\gamma} is modeled as a unitary system, then 此外,集合 {O_(alpha gamma)}\left\{O_{\alpha \gamma}\right\} 是一个 POVM。如果 Q_(gamma)Q_{\gamma} 被建模为一个单位系统,则
The set O_(gamma)O_{\gamma} is still a POVM with a single element, K_(gamma)=U_(gamma)K_{\gamma}=U_{\gamma}. 集合 O_(gamma)O_{\gamma} 仍然是一个具有单个元素 K_(gamma)=U_(gamma)K_{\gamma}=U_{\gamma} 的 POVM。
Figure 31.12 shows a schematic representation of an experimental setup for QST performed in the Clarendon Lab at Oxford. In this case, for the probability outcomes p_(alpha gamma)p_{\alpha \gamma}, the outcomes alpha in{0,1}\alpha \in\{0,1\} and the configurations gamma in{Omega,T}\gamma \in\{\Omega, T\} were selected from frequency and time of the fluoresence signal [80]. 图 31.12 显示了在牛津的克拉伦登实验室进行 QST 的实验设置的示意图。在这种情况下,对于概率结果 p_(alpha gamma)p_{\alpha \gamma} ,结果 alpha in{0,1}\alpha \in\{0,1\} 和配置 gamma in{Omega,T}\gamma \in\{\Omega, T\} 是从荧光信号的频率和时间中选择的[80]。
31.4.1.2 Maximum Likelihood 31.4.1.2 最大似然估计
The ML approach to quantum state estimation presented in this section, as well as observing that the estimation is convex, can be found in [81,82][81,82] and the references therein. Using convex programming methods, such as an interior-point algorithm for computation, was not exploited in these references. 本节中提出的量子态估计的机器学习方法,以及观察到估计是凸的,可以在 [81,82][81,82] 及其中的参考文献中找到。这些参考文献中并未利用凸编程方法,例如用于计算的内点算法。
If the experiments are independent, then the probability of obtaining the data (Equation 3.14) is a product of the individual model probabilities (Equation 3.16). Consequently, for an assumed initial state rho\rho, the model predicts that the probability of obtaining the data set (Equation 3.14) is given by, P{D,rho}=\mathbf{P}\{D, \rho\}=prod_(alpha,gamma)p_(alpha gamma)(rho)^(n_(alpha)gamma)\prod_{\alpha, \gamma} p_{\alpha \gamma}(\rho)^{n_{\alpha} \gamma}. The data are thus captured in the outcome counts {n_(alpha gamma)}\left\{n_{\alpha \gamma}\right\}, whereas the model terms have a rho\rho-dependence. The ML estimate of rho\rho is obtained by finding a rho\rho in the set (Equation 3.17) which maximizes 如果实验是独立的,那么获得数据的概率(方程 3.14)是各个模型概率(方程 3.16)的乘积。因此,对于假设的初始状态 rho\rho ,模型预测获得数据集的概率(方程 3.14)由 P{D,rho}=\mathbf{P}\{D, \rho\}=prod_(alpha,gamma)p_(alpha gamma)(rho)^(n_(alpha)gamma)\prod_{\alpha, \gamma} p_{\alpha \gamma}(\rho)^{n_{\alpha} \gamma} 给出。因此,数据被捕获在结果计数 {n_(alpha gamma)}\left\{n_{\alpha \gamma}\right\} 中,而模型项具有 rho\rho -依赖性。 rho\rho 的最大似然估计通过在集合中找到一个 rho\rho (方程 3.17)来获得,该 rho\rho 最大化。
Diatomic molecular system 双原子分子系统
Electronic transition 电子跃迁
Signal from fluorescence measurements of vibrations 来自荧光测量振动的信号
FIGURE 31.12 QST in the lab. State tomography of vibrational wavepackets in diatomic molecules. 图 31.12 实验室中的 QST。二原子分子中振动波包的状态层析。
the P{D,rho}\mathbf{P}\{D, \rho\}, or equivalently, minimizes the negative log\log-likelihood function, L(D,rho)=-log P{D,rho}L(D, \rho)=-\log \mathbf{P}\{D, \rho\}. The ML state estimate, rho^(ML)\rho^{\mathrm{ML}}, is obtained as the solution to the optimization problem P{D,rho}\mathbf{P}\{D, \rho\} ,或等价地,最小化负 log\log -似然函数 L(D,rho)=-log P{D,rho}L(D, \rho)=-\log \mathbf{P}\{D, \rho\} 。最大似然状态估计 rho^(ML)\rho^{\mathrm{ML}} 作为优化问题的解得出。
L(D,rho)L(D, \rho) is a positively weighted sum of log-convex functions of rho\rho, and hence, is a log-convex function of rho\rho. The constraint that rho\rho is a density matrix forms a convex set in rho\rho. Hence, Equation 31.20 is in a category of a class of well-studied log-convex optimization problems, for example, [49]. L(D,rho)L(D, \rho) 是 rho\rho 的对数凸函数的正权重和,因此是 rho\rho 的对数凸函数。约束条件 rho\rho 是一个密度矩阵,形成了 rho\rho 中的一个凸集。因此,方程 31.20 属于一类研究较多的对数凸优化问题,例如 [49]。
31.4.1.3 Least Squares 31.4.1.3 最小二乘法
In a typical application, the number of trials per configuration, ℓ_(gamma)\ell_{\gamma}, is sufficiently large so that the empirical estimate of the outcome probability, is a good estimate of the true outcome probability p_(alpha gamma)^("true ")p_{\alpha \gamma}^{\text {true }} 在典型应用中,每个配置的试验次数 ℓ_(gamma)\ell_{\gamma} 足够大,以至于结果概率的经验估计是对真实结果概率 p_(alpha gamma)^("true ")p_{\alpha \gamma}^{\text {true }} 的良好估计
This leads to the least squares (LS) state estimate rho^(LS)\rho^{\mathrm{LS}} as the solution to the constrained weighted LS problem 这导致最小二乘(LS)状态估计 rho^(LS)\rho^{\mathrm{LS}} 作为约束加权最小二乘问题的解
Is is often the case that the process matrix is sparse or almost sparse, that is, it consists of a small number of significant elements. In some cases, a known sparsity pattern can arise from the underlying dynamics, thereby inherently increasing QPT efficiency [85]. 过程矩阵通常是稀疏或几乎稀疏的,即它由少量显著元素组成。在某些情况下,已知的稀疏模式可以源于潜在的动态,从而固有地提高量子过程 tomography(QPT)的效率 [85]。 In most cases, however, the sparsity pattern is not known. In this more common case we can apply the methods of CS [53-55]. Specifically, for a class of incomplete linear measurement equations ( y=Ax,A inR^(m xx N),m≪Ny=A x, A \in \mathbf{R}^{m \times N}, m \ll N ), constrained ℓ_(1)\ell_{1}-norm minimization (minimize ||x||_(ℓ_(1))\|x\|_{\ell_{1}} subject to y=Axy=A x ), a convex optimization problem can perfectly estimate the sparse variable xx. These methods are also robust to measurement noise and for almost sparse variables. 然而,在大多数情况下,稀疏模式是未知的。在这种更常见的情况下,我们可以应用压缩感知的方法[53-55]。具体来说,对于一类不完整的线性测量方程( y=Ax,A inR^(m xx N),m≪Ny=A x, A \in \mathbf{R}^{m \times N}, m \ll N ),约束 ℓ_(1)\ell_{1} -范数最小化(最小化 ||x||_(ℓ_(1))\|x\|_{\ell_{1}} ,满足 y=Axy=A x ),一个凸优化问题可以完美地估计稀疏变量 xx 。这些方法对测量噪声也具有鲁棒性,并且适用于几乎稀疏的变量。
For a quantum information system the ideal quantum logic gates are unitaries, that is, Q(rho)=U rhoU^(†)Q(\rho)=U \rho U^{\dagger}. Let { bar(B)_(alpha)inC^(n^(2)xxn^(2)),alpha=1,dots,n^(2)}\left\{\bar{B}_{\alpha} \in \mathbf{C}^{n^{2} \times n^{2}}, \alpha=1, \ldots, n^{2}\right\} denote the “Natural-Basis,” that is, each basis matrix has a single nonzero element of one. In this basis, the process matrix associated with the ideal unitary channel has the rank-1 form, X_("nat ")=xx^(†)X_{\text {nat }}=x x^{\dagger} with x inC^(n^(2)),x^(†)x=nx \in \mathbf{C}^{n^{2}}, x^{\dagger} x=n. A SVD gives X_("nat ")=V diag(n,0,dots,0)V^(†)X_{\text {nat }}=V \operatorname{diag}(n, 0, \ldots, 0) V^{\dagger} with V inC^(n^(2)xxn^(2))V \in \mathbf{C}^{n^{2} \times n^{2}} a 对于量子信息系统,理想的量子逻辑门是单位算子,即 Q(rho)=U rhoU^(†)Q(\rho)=U \rho U^{\dagger} 。让 { bar(B)_(alpha)inC^(n^(2)xxn^(2)),alpha=1,dots,n^(2)}\left\{\bar{B}_{\alpha} \in \mathbf{C}^{n^{2} \times n^{2}}, \alpha=1, \ldots, n^{2}\right\} 表示“自然基”,即每个基矩阵只有一个非零元素为一。在这个基中,与理想单位通道相关的过程矩阵具有秩为 1 的形式 X_("nat ")=xx^(†)X_{\text {nat }}=x x^{\dagger} ,并且 x inC^(n^(2)),x^(†)x=nx \in \mathbf{C}^{n^{2}}, x^{\dagger} x=n 。奇异值分解(SVD)给出 X_("nat ")=V diag(n,0,dots,0)V^(†)X_{\text {nat }}=V \operatorname{diag}(n, 0, \ldots, 0) V^{\dagger} ,并且 V inC^(n^(2)xxn^(2))V \in \mathbf{C}^{n^{2} \times n^{2}} 。
unitary. An equivalent process matrix can be formed from the SVD in what is referred to here as the “SVD-Basis,” {B_(alpha)=sum_(alpha^(')=1)^(n_(S)^(2))V_(alpha^(')alpha) bar(B)_(alpha^(')),alpha=1,dots,n^(2)}\left\{B_{\alpha}=\sum_{\alpha^{\prime}=1}^{n_{S}^{2}} V_{\alpha^{\prime} \alpha} \bar{B}_{\alpha^{\prime}}, \alpha=1, \ldots, n^{2}\right\}. The equivalent process matrix for a unitary channel, in this basis, denoted by X_("svd ")X_{\text {svd }}, is maximally sparse with a single nonzero element, specifically, (X_("nat "))_(11)=n\left(X_{\text {nat }}\right)_{11}=n. The actual channel which interacts with an environment will be a perturbation of the ideal unitary. If the noise source is small, then the process matrix in the nominal basis will be almost sparse. 单位。可以从奇异值分解(SVD)形成一个等效的过程矩阵,这里称为“SVD 基”, {B_(alpha)=sum_(alpha^(')=1)^(n_(S)^(2))V_(alpha^(')alpha) bar(B)_(alpha^(')),alpha=1,dots,n^(2)}\left\{B_{\alpha}=\sum_{\alpha^{\prime}=1}^{n_{S}^{2}} V_{\alpha^{\prime} \alpha} \bar{B}_{\alpha^{\prime}}, \alpha=1, \ldots, n^{2}\right\} 。在这个基中,单位通道的等效过程矩阵用 X_("svd ")X_{\text {svd }} 表示,具有最大稀疏性,只有一个非零元素,具体为 (X_("nat "))_(11)=n\left(X_{\text {nat }}\right)_{11}=n 。与环境相互作用的实际通道将是理想单位的一个扰动。如果噪声源很小,那么名义基中的过程矩阵将几乎是稀疏的。
As an example, consider a system which is ideally a two-qubit (n=4)(n=4) quantum memory, thus U=I_(4)U=I_{4}. Suppose the actual system is a perturbation of identity by independent bit-flip errors in each channel occurring with probability p_(bf)p_{\mathrm{bf}}. For p_(bf)=0.05p_{\mathrm{bf}}=0.05 and p_(bf)=0.2p_{\mathrm{bf}}=0.2, the respective channel fidelities are about 0.90 and 0.64 , which for quantum information processing would need to be discovered by QPT and then corrected for the device to ever work. Referring to Figure 31.17, in the Natural-Basis, the ideal 16 xx1616 \times 16 process matrix has 16 nonzero elements out of 256 , all of magnitude one. Using the SVD-Basis, the corresponding process matrix as shown in Figure 31.17b has a single nonzero element of magnitude 作为一个例子,考虑一个理想的两量子比特 (n=4)(n=4) 量子存储系统,因此 U=I_(4)U=I_{4} 。假设实际系统是通过每个通道中以概率 p_(bf)p_{\mathrm{bf}} 发生的独立比特翻转错误对身份的扰动。对于 p_(bf)=0.05p_{\mathrm{bf}}=0.05 和 p_(bf)=0.2p_{\mathrm{bf}}=0.2 ,相应的通道保真度约为 0.90 和 0.64,这对于量子信息处理需要通过量子过程 tomography(QPT)发现,然后进行纠正,以使设备能够正常工作。参考图 31.17,在自然基中,理想的 16 xx1616 \times 16 过程矩阵在 256 个元素中有 16 个非零元素,所有元素的大小为 1。使用 SVD 基,图 31.17b 所示的相应过程矩阵具有一个大小为的单个非零元素。
where the control design function bar(epsi)( widehat(omega)_(1)^((i)))\bar{\varepsilon}\left(\widehat{\omega}_{1}^{(i)}\right) represents the pulse design from the above table (Figure 31.20), and where the average likelihood function follows from the description in Section 31.4.3.1 with the following parameters: 控制设计函数 bar(epsi)( widehat(omega)_(1)^((i)))\bar{\varepsilon}\left(\widehat{\omega}_{1}^{(i)}\right) 代表上述表格(图 31.20)中的脉冲设计,平均似然函数则根据第 31.4.3.1 节的描述得出,具有以下参数:
Single initial state rho^("init ")=|0:)(:0|quad(beta=1)
POVM M_(1)=|0:)(:0|,quadM_(2)=|1:)(:1|quad(n_("out ")=2)
Sample times either {t_(f)( widehat(omega)_(1)),quad(n_("sa ")=1)} or {t_(f)( widehat(omega)_(1))//2,quadt_(f)( widehat(omega)_(1)),(n_("sa ")=2)}.| Single initial state | $\rho^{\text {init }}=\|0\rangle\langle 0\| \quad(\beta=1)$ |
| :--- | :--- |
| POVM | $M_{1}=\|0\rangle\langle 0\|, \quad M_{2}=\|1\rangle\langle 1\| \quad\left(n_{\text {out }}=2\right)$ |
| Sample times | either $\left\{t_{\mathrm{f}}\left(\widehat{\omega}_{1}\right), \quad\left(n_{\text {sa }}=1\right)\right\}$ or $\left\{t_{\mathrm{f}}\left(\widehat{\omega}_{1}\right) / 2, \quad t_{\mathrm{f}}\left(\widehat{\omega}_{1}\right),\left(n_{\text {sa }}=2\right)\right\}$. |
Using Hamiltonian parameters (omega_(0)^("true ")=1,omega_(1)^("true ")=0.01,omega_(c)^("true ")=0.01)\left(\omega_{0}^{\text {true }}=1, \omega_{1}^{\text {true }}=0.01, \omega_{c}^{\text {true }}=0.01\right), Figure 31.21 shows PL( widehat(omega)_(1):}\mathbb{P} L\left(\widehat{\omega}_{1}\right., n_(sa)=1n_{\mathrm{sa}}=1 ) versus widehat(omega)_(1)//omega_(i)^("true ")\widehat{\omega}_{1} / \omega_{i}^{\text {true }} for sequences of adaptive iterations using the estimate widehat(omega)_(1)\widehat{\omega}_{1} obtained from a local hill-climbing algorithm, that is, the local maximum of the average likelihood function is obtained. The estimation is followed by a control using the estimated value in the pulse control table. 使用哈密顿参数 (omega_(0)^("true ")=1,omega_(1)^("true ")=0.01,omega_(c)^("true ")=0.01)\left(\omega_{0}^{\text {true }}=1, \omega_{1}^{\text {true }}=0.01, \omega_{c}^{\text {true }}=0.01\right) ,图 31.21 显示了 PL( widehat(omega)_(1):}\mathbb{P} L\left(\widehat{\omega}_{1}\right. , n_(sa)=1n_{\mathrm{sa}}=1 ) 相对于 widehat(omega)_(1)//omega_(i)^("true ")\widehat{\omega}_{1} / \omega_{i}^{\text {true }} 的自适应迭代序列,使用从局部爬山算法获得的估计 widehat(omega)_(1)\widehat{\omega}_{1} ,即获得平均似然函数的局部最大值。估计之后,使用脉冲控制表中的估计值进行控制。 In the two cases shown, the algorithm converges to the true value. 在这两个案例中,算法收敛到真实值。
Although not shown, the algorithm does not converge from all initial values of widehat(omega)_(1)\widehat{\omega}_{1}. Figure 31.21c shows ||U(t_(f)( widehat(omega)_(1)),( bar(epsi))( widehat(omega)_(1)))-U_("des ")||_("frob ")\left\|U\left(t_{\mathrm{f}}\left(\widehat{\omega}_{1}\right), \bar{\varepsilon}\left(\widehat{\omega}_{1}\right)\right)-U_{\text {des }}\right\|_{\text {frob }} versus estimate widehat(omega)_(1)//omega_(1)^("true ")\widehat{\omega}_{1} / \omega_{1}^{\text {true }} with the control from the table. The function is clearly not convex. The region of convergence for n_(sa)=1n_{\mathrm{sa}}=1 is 0.9 <= widehat(omega)_(1)//omega_(1)^("true ") <= 1.30.9 \leq \widehat{\omega}_{1} / \omega_{1}^{\text {true }} \leq 1.3, for n_(sa)=2,0.6 <=n_{\mathrm{sa}}=2,0.6 \leqwidehat(omega)_(1)//omega_(1)^("true ") <= 2.5\widehat{\omega}_{1} / \omega_{1}^{\text {true }} \leq 2.5. In neither case will the algorithm converge for widehat(omega)_(1)//omega_(1)^("true ") < 0.6\widehat{\omega}_{1} / \omega_{1}^{\text {true }}<0.6. These results, of course, are specific to this example and cannot be generalized. To reiterate, conditions for convergence, region of attraction, and so on, are only partially understood, in general, for this type of iteration [91]. 尽管未显示,该算法并不从所有初始值 widehat(omega)_(1)\widehat{\omega}_{1} 收敛。图 31.21c 显示了 ||U(t_(f)( widehat(omega)_(1)),( bar(epsi))( widehat(omega)_(1)))-U_("des ")||_("frob ")\left\|U\left(t_{\mathrm{f}}\left(\widehat{\omega}_{1}\right), \bar{\varepsilon}\left(\widehat{\omega}_{1}\right)\right)-U_{\text {des }}\right\|_{\text {frob }} 与估计 widehat(omega)_(1)//omega_(1)^("true ")\widehat{\omega}_{1} / \omega_{1}^{\text {true }} 的关系,控制来自表格。该函数显然不是凸的。 n_(sa)=1n_{\mathrm{sa}}=1 的收敛区域是 0.9 <= widehat(omega)_(1)//omega_(1)^("true ") <= 1.30.9 \leq \widehat{\omega}_{1} / \omega_{1}^{\text {true }} \leq 1.3 ,对于 n_(sa)=2,0.6 <=n_{\mathrm{sa}}=2,0.6 \leqwidehat(omega)_(1)//omega_(1)^("true ") <= 2.5\widehat{\omega}_{1} / \omega_{1}^{\text {true }} \leq 2.5 。在这两种情况下,算法都不会收敛到 widehat(omega)_(1)//omega_(1)^("true ") < 0.6\widehat{\omega}_{1} / \omega_{1}^{\text {true }}<0.6 。这些结果当然是特定于此示例的,不能被推广。重申一下,收敛条件、吸引区域等,对于这种类型的迭代一般来说只部分被理解[91]。
31.5 Optimal Quantum Feedback Control 31.5 最优量子反馈控制
在方程中出现的矩阵不是任意的——它们必须满足某些约束条件才能描述量子系统。具体来说,我们考虑描述开放谐振子组装的量子线性系统,例如腔体的互连。 These systems are described by quantum stochastic differential equations with quantum ww and classical uu inputs, and an output yy available to the controller 这些系统由量子随机微分方程描述,具有量子 ww 和经典 uu 输入,以及可供控制器使用的输出 yy
where A,B_(1),B_(2),CA, B_{1}, B_{2}, C, and DD are, respectively, real 2n xx2n,2n xx2p,2n xx m,2p xx2n2 n \times 2 n, 2 n \times 2 p, 2 n \times m, 2 p \times 2 n, and 2p xx2p2 p \times 2 p matrices, and x(t)=(x_(1)(t),dots,x_(2n)(t))^(T)x(t)=\left(x_{1}(t), \ldots, x_{2 n}(t)\right)^{T} is a vector of system variables. The initial system variables x(0)=xx(0)=x are 其中 A,B_(1),B_(2),CA, B_{1}, B_{2}, C 和 DD 分别是真实的 2n xx2n,2n xx2p,2n xx m,2p xx2n2 n \times 2 n, 2 n \times 2 p, 2 n \times m, 2 p \times 2 n 和 2p xx2p2 p \times 2 p 矩阵,而 x(t)=(x_(1)(t),dots,x_(2n)(t))^(T)x(t)=\left(x_{1}(t), \ldots, x_{2 n}(t)\right)^{T} 是系统变量的向量。初始系统变量 x(0)=xx(0)=x 为
where I_(n)I_{n} is the n xx nn \times n identity matrix. The signal w(t)w(t) is a vector of self-adjoint stochastic processes with covariance 其中 I_(n)I_{n} 是 n xx nn \times n 单位矩阵。信号 w(t)w(t) 是具有协方差的自伴随机过程的向量。
where FF is the nonnegative Hermitian matrix F=I_(2p)+iJ_(2p)F=I_{2 p}+i J_{2 p}. The input u(t)u(t) is a vector of classical processes (self-adjoint). 其中 FF 是非负厄米矩阵 F=I_(2p)+iJ_(2p)F=I_{2 p}+i J_{2 p} 。输入 u(t)u(t) 是经典过程的向量(自伴随)。
As mentioned above, the linear system (Equation 31.55) looks formally like a classical linear system. The essential difference is that the vectors x(t),w(t)x(t), w(t), and y(t)y(t) have components that do not commute. For arbitrary matrices A,B_(1),B_(2),CA, B_{1}, B_{2}, C, and DD, the linear system (Equation 31.55) need not necessarily correspond to a quantum system, and so it is important to know when Equation 31.55 does in fact describe a quantum physical system. 如上所述,线性系统(方程 31.55)在形式上看起来像一个经典线性系统。根本的区别在于向量 x(t),w(t)x(t), w(t) 和 y(t)y(t) 的分量不满足交换律。对于任意矩阵 A,B_(1),B_(2),CA, B_{1}, B_{2}, C 和 DD ,线性系统(方程 31.55)不一定对应于量子系统,因此了解方程 31.55 何时确实描述量子物理系统是很重要的。
The open quantum systems that give rise to Heisenberg equations of the type (Equation 31.55) are those for which the SS-system (recall Section 31.2.3) is a collection of harmonic oscillators (Section 31.2.5), and the EE-system is the free field (Section 31.2.6). The unitary dynamics of the SES E-system means that we require x(t)=U^(†)(t)xU(t)x(t)=U^{\dagger}(t) x U(t) and y(t)=U^(†)(t)w(t)U(t)y(t)=U^{\dagger}(t) w(t) U(t), where U(t)=U_(SE)(t)U(t)=U_{S E}(t) is a unitary operator solving Schrodinger’s equation (in Stratonovich form) 产生海森堡方程(方程 31.55)的开放量子系统是那些 SS -系统(回顾第 31.2.3 节)是谐振子集合(第 31.2.5 节),而 EE -系统是自由场(第 31.2.6 节)。 SES E -系统的单位动力学意味着我们需要 x(t)=U^(†)(t)xU(t)x(t)=U^{\dagger}(t) x U(t) 和 y(t)=U^(†)(t)w(t)U(t)y(t)=U^{\dagger}(t) w(t) U(t) ,其中 U(t)=U_(SE)(t)U(t)=U_{S E}(t) 是解决薛定谔方程(在斯特拉托诺维奇形式下)的单位算符。
where w(t)=(b_(in,r)^(T)(t),b_(in,i)^(T)(t))^(T),L=Mx,H=(1)/(2)(x^(T)Rx+x^(T)Su+u^(T)S^(T)x)w(t)=\left(b_{i n, r}^{T}(t), b_{i n, i}^{T}(t)\right)^{T}, L=M x, H=\frac{1}{2}\left(x^{T} R x+x^{T} S u+u^{T} S^{T} x\right), and RR is real symmetric and SS is real. If the system is specified by these parameters, then the matrices A,B_(1),B_(2),CA, B_{1}, B_{2}, C, and DD are given by 在 w(t)=(b_(in,r)^(T)(t),b_(in,i)^(T)(t))^(T),L=Mx,H=(1)/(2)(x^(T)Rx+x^(T)Su+u^(T)S^(T)x)w(t)=\left(b_{i n, r}^{T}(t), b_{i n, i}^{T}(t)\right)^{T}, L=M x, H=\frac{1}{2}\left(x^{T} R x+x^{T} S u+u^{T} S^{T} x\right) 的地方, RR 是实对称的, SS 是实数。如果系统由这些参数指定,则矩阵 A,B_(1),B_(2),CA, B_{1}, B_{2}, C 和 DD 由以下公式给出:
and D=ID=I. 和 D=ID=I 。
As alluded to in Sections 31.2.5 and 31.2.7 [25], Equations 31.55 describe a quantum physical system in this sense if and only if the matrices A,B_(1),B_(2),CA, B_{1}, B_{2}, C, and DD are real and satisfy the following conditions: 如第 31.2.5 节和第 31.2.7 节所提到的[25],方程 31.55 在这个意义上描述一个量子物理系统当且仅当矩阵 A,B_(1),B_(2),CA, B_{1}, B_{2}, C 和 DD 是实数并满足以下条件:
Indeed, if A,B_(1),B_(2),CA, B_{1}, B_{2}, C, and DD satisfy these conditions, then R=(1)/(2)(-J_(n)A+A^(T)J_(n)),M=(1)/(2)[IiI]CR=\frac{1}{2}\left(-J_{n} A+A^{T} J_{n}\right), M=\frac{1}{2}[I i I] C, and S=-J_(n)B_(2)S=-J_{n} B_{2}. 确实,如果 A,B_(1),B_(2),CA, B_{1}, B_{2}, C 和 DD 满足这些条件,那么 R=(1)/(2)(-J_(n)A+A^(T)J_(n)),M=(1)/(2)[IiI]CR=\frac{1}{2}\left(-J_{n} A+A^{T} J_{n}\right), M=\frac{1}{2}[I i I] C 和 S=-J_(n)B_(2)S=-J_{n} B_{2} 。
The optical cavity described in Section 31.2.7 is an example of a linear quantum system. 在第 31.2.7 节中描述的光学腔是线性量子系统的一个例子。
31.5.2 Quantum Filtering 31.5.2 量子滤波
It is often of interest to monitor an observable y_(0)(t)y_{0}(t) of the output field y(t)y(t), as this information may be used to make inferences about the system, or for controlling the system, see Figure 31.22. 通常会对输出场 y(t)y(t) 的可观察量 y_(0)(t)y_{0}(t) 进行监测,因为这些信息可以用来推断系统或控制系统,见图 31.22。
FIGURE 31.22 Filtering a measurement signal y_(0)(*)y_{0}(\cdot). 图 31.22 过滤测量信号 y_(0)(*)y_{0}(\cdot) 。
In quantum optics, such measurements are made using photodetection systems; for instance, homodyne detection is used to (approximately) measure the real quadrature y_(0)(t)=[I_(p)0]y(t)y_{0}(t)=\left[I_{p} 0\right] y(t), the photocurrent produced by the detector. Note that [y_(0,j)(s),y_(0,k)(t)]=0\left[y_{0, j}(s), y_{0, k}(t)\right]=0 for all j,kj, k and times s,ts, t, and so the components of y_(0)(s),0 <= s <= ty_{0}(s), 0 \leq s \leq t form a commuting collection of observables. The measurement signal is given by 在量子光学中,这种测量是通过光电探测系统进行的;例如,同相检测用于(近似)测量实象量 y_(0)(t)=[I_(p)0]y(t)y_{0}(t)=\left[I_{p} 0\right] y(t) ,即探测器产生的光电流。注意 [y_(0,j)(s),y_(0,k)(t)]=0\left[y_{0, j}(s), y_{0, k}(t)\right]=0 对于所有 j,kj, k 和时间 s,ts, t ,因此 y_(0)(s),0 <= s <= ty_{0}(s), 0 \leq s \leq t 的分量形成一个可交换的可观测量集合。测量信号由以下公式给出:
where C_(0)=[I_(p)0]CC_{0}=\left[I_{p} 0\right] C and w_(0)(t)=[I_(p)0]w(t)w_{0}(t)=\left[I_{p} 0\right] w(t) is a standard classical Wiener process. 其中 C_(0)=[I_(p)0]CC_{0}=\left[I_{p} 0\right] C 和 w_(0)(t)=[I_(p)0]w(t)w_{0}(t)=\left[I_{p} 0\right] w(t) 是标准经典维纳过程。
量子滤波器计算量子系统的条件状态 pi_(t)\pi_{t} ,它是为比这里考虑的线性系统更一般的开放量子系统开发的,可以视为经典非线性滤波方程的量子推广。 In the case of linear gaussian quantum systems, the quantum filter reduces to a Kalman filter. 在线性高斯量子系统的情况下,量子滤波器简化为卡尔曼滤波器。
Now x(t)x(t) commutes with y_(0)(s),0 <= s <= ty_{0}(s), 0 \leq s \leq t, and so the quantum conditional expectation widehat(x)(t)=\widehat{x}(t)=P[x(t)∣y_(0)(s),0 <= s <= t]=Tr[pi_(t)x]\mathbb{P}\left[x(t) \mid y_{0}(s), 0 \leq s \leq t\right]=\operatorname{Tr}\left[\pi_{t} x\right] is well defined, [12,13,42,75]. The time evolution of widehat(x)(t)\widehat{x}(t) is given by the quantum filter 现在 x(t)x(t) 与 y_(0)(s),0 <= s <= ty_{0}(s), 0 \leq s \leq t 通勤,因此量子条件期望 widehat(x)(t)=\widehat{x}(t)=P[x(t)∣y_(0)(s),0 <= s <= t]=Tr[pi_(t)x]\mathbb{P}\left[x(t) \mid y_{0}(s), 0 \leq s \leq t\right]=\operatorname{Tr}\left[\pi_{t} x\right] 是明确定义的,[12,13,42,75]。 widehat(x)(t)\widehat{x}(t) 的时间演化由量子滤波器给出。
The conditional mean widehat(x)(t)\widehat{x}(t) and symmetrized conditional covariance Y(t)=(1)/(2)P[x(t)x^(T)(t)+(x(t)x^(T)(t))^(T)]Y(t)=\frac{1}{2} \mathbb{P}\left[x(t) x^{T}(t)+\left(x(t) x^{T}(t)\right)^{T}\right] parameterize the conditional state pi_(t)\pi_{t}. 条件均值 widehat(x)(t)\widehat{x}(t) 和对称条件协方差 Y(t)=(1)/(2)P[x(t)x^(T)(t)+(x(t)x^(T)(t))^(T)]Y(t)=\frac{1}{2} \mathbb{P}\left[x(t) x^{T}(t)+\left(x(t) x^{T}(t)\right)^{T}\right] 参数化条件状态 pi_(t)\pi_{t} 。
Example: Monitoring an Atom in a Cavity 示例:监测腔体中的原子
Consider the problem of measurement feedback control of an atom trapped inside an optical cavity, as shown in Figure 31.7 [17]. 考虑光学腔内被捕获原子的测量反馈控制问题,如图 31.7 所示[17]。 A magnetic field and a second optical beam create a spatially dependent force within the cavity which provides a trapping mechanism which can be adjusted by classical control signals. 一个磁场和第二束光束在腔体内产生一个空间依赖的力,这提供了一种可以通过经典控制信号调整的捕获机制。 The control objective is to cool and confines the atom, that is, to reduce the atom’s momentum and to spatially localize the atom. Here, we present the quantum filter for this problem, and look at control aspects in the following sections. 控制目标是冷却并限制原子,即减少原子的动量并在空间上局部化原子。在这里,我们提出了针对这个问题的量子滤波器,并在接下来的章节中讨论控制方面。
A linear quadratic Gaussian (LQG) approximation gives a linear quantum model for the trapped atom with parameters 线性二次高斯(LQG)近似为被捕获的原子提供了一个线性量子模型,参数为
determining the Hamiltonian and coupling operator (recall Section 31.5.1). Here, mm is the mass of the atom, omega\omega is the natural frequency, and kk is a coupling strength parameter. The matrices A,B_(1),B_(2)A, B_{1}, B_{2}, CC, and DD are given by 确定哈密顿量和耦合算符(回顾第 31.5.1 节)。这里, mm 是原子的质量, omega\omega 是自然频率, kk 是耦合强度参数。矩阵 A,B_(1),B_(2)A, B_{1}, B_{2} 、 CC 和 DD 由以下给出:
The vector x=(q,p)^(T)x=(q, p)^{T} is formed from the position qq and momentum pp observables for the atom. It can be seen that the uncontrolled mean motion is oscillatory. 向量 x=(q,p)^(T)x=(q, p)^{T} 是由原子的位置信息 qq 和动量信息 pp 形成的。可以看出,无控制的平均运动是振荡的。
The measurement signal y_(0)(*)y_{0}(\cdot) is given by Equation 31.64, where C_(0)=[[,0]]C=[2sqrt(2k)0]C_{0}=\left[\begin{array}{ll} & 0\end{array}\right] C=[2 \sqrt{2 k} 0]. The vector of estimates widehat(x)=[ widehat(q), widehat(p)]^(T)\widehat{x}=[\widehat{q}, \widehat{p}]^{T} evolves according to the quantum (Kalman) filter 测量信号 y_(0)(*)y_{0}(\cdot) 由方程 31.64 给出,其中 C_(0)=[[,0]]C=[2sqrt(2k)0]C_{0}=\left[\begin{array}{ll} & 0\end{array}\right] C=[2 \sqrt{2 k} 0] 。估计向量 widehat(x)=[ widehat(q), widehat(p)]^(T)\widehat{x}=[\widehat{q}, \widehat{p}]^{T} 根据量子(卡尔曼)滤波器演变。
{:[Y^(˙)_(11)(t)=(2Y_(12)(t))/(m)-8kY_(11)^(2)(t)],[Y^(˙)_(12)(t)=(Y_(22)(t))/(m)-momega^(2)Y_(11)(t)-8kY_(11)(t)Y_(12)(t)],[Y^(˙)_(22)(t)=-2momega^(2)Y_(12)(t)+2k-8kY_(12)^(2)(t)]:}\begin{aligned}
& \dot{Y}_{11}(t)=\frac{2 Y_{12}(t)}{m}-8 k Y_{11}^{2}(t) \\
& \dot{Y}_{12}(t)=\frac{Y_{22}(t)}{m}-m \omega^{2} Y_{11}(t)-8 k Y_{11}(t) Y_{12}(t) \\
& \dot{Y}_{22}(t)=-2 m \omega^{2} Y_{12}(t)+2 k-8 k Y_{12}^{2}(t)
\end{aligned}
31.5.3 Quantum Measurement Feedback LQG Control 31.5.3 量子测量反馈 LQG 控制
Kalman’s LQG control is one of the most significant developments in control theory [92]. In this section, we explain how measurement feedback LQG [measurement feedback (MF)-quantum linear quadratic Gaussian (QLQG)] works for the quantum linear system (Equation 31.55) [17,75]. 卡尔曼的 LQG 控制是控制理论中最重要的发展之一[92]。在本节中,我们将解释测量反馈 LQG [测量反馈(MF)-量子线性二次高斯(QLQG)]如何在量子线性系统(方程 31.55)中工作[17,75]。 An experimental demonstration of the application of LQG control in quantum optics is described in [20]. 在[20]中描述了 LQG 控制在量子光学中应用的实验演示。
We begin by stating what we mean by a measurement feedback controller KK for the quantum linear system (Equation 31.55). Such a controller KK is to process the measurement signal y_(0)(*)y_{0}(\cdot) (discussed in Section 31.5.2) in a causal manner to produce the classical control signal u(*)u(\cdot). Thus, for each time t >= 0t \geq 0, u(t)=K_(t)(y_(0)(s),0 <= s <= t)u(t)=K_{t}\left(y_{0}(s), 0 \leq s \leq t\right), Figure 31.23 . 我们首先说明我们所指的量测反馈控制器 KK 对于量子线性系统(方程 31.55)。这样的控制器 KK 以因果方式处理量测信号 y_(0)(*)y_{0}(\cdot) (在第 31.5.2 节中讨论)以产生经典控制信号 u(*)u(\cdot) 。因此,对于每个时间 t >= 0t \geq 0 , u(t)=K_(t)(y_(0)(s),0 <= s <= t)u(t)=K_{t}\left(y_{0}(s), 0 \leq s \leq t\right) ,见图 31.23。
LQG 性能标准由以下定义
J(K)=P[int_(0)^(T)[(1)/(2)x^(T)(t)Px(t)+(1)/(2)u^(T)(t)Qu(t)]dt+(1)/(2)x^(T)(T)X_(T)x(T)]J(K)=\mathbb{P}\left[\int_{0}^{T}\left[\frac{1}{2} x^{T}(t) P x(t)+\frac{1}{2} u^{T}(t) Q u(t)\right] d t+\frac{1}{2} x^{T}(T) X_{T} x(T)\right]
为了解决这个问题,我们回忆起量子条件期望 P[x^(T)(t)Px(t)∣y_(0)(s),0 <= :}\mathbb{P}\left[x^{T}(t) P x(t) \mid y_{0}(s), 0 \leq\right.s <= t]s \leq t] 是明确定义的,因为 x(t)x(t) 与可交换的可观测量家族 y_(0)(s),0 <= s <= ty_{0}(s), 0 \leq s \leq t 是可交换的,
图 31.23 量子线性系统的测量反馈控制。
而且,满足基本属性 P[P[x^(T)(t)Px(t)∣y_(0)(s),0 <= s <= t]]=P[x^(T)(t)Px(t)]\mathbb{P}\left[\mathbb{P}\left[x^{T}(t) P x(t) \mid y_{0}(s), 0 \leq s \leq t\right]\right]=\mathbb{P}\left[x^{T}(t) P x(t)\right] 。此外, P[x^(T)(t)Px(t)∣y_(0)(s),0 <= s <= t]= widehat(x)^(T)(t)P widehat(x)(t)+Tr[PY(t)]\mathbb{P}\left[x^{T}(t) P x(t) \mid y_{0}(s), 0 \leq s \leq t\right]=\widehat{x}^{T}(t) P \widehat{x}(t)+\operatorname{Tr}[P Y(t)] 。我们可以将 J(K)J(K) 重新表达为一个等效的经典控制问题,涉及量子滤波器:
J(K)=P[int_(0)^(T)[(1)/(2) widehat(x)^(T)(t)P( widehat(x))(t)+(1)/(2)u^(T)(t)Qu(t)]dt+(1)/(2) widehat(x)^(T)(T)X_(T)( widehat(x))(T)]+alphaJ(K)=\mathbf{P}\left[\int_{0}^{T}\left[\frac{1}{2} \widehat{x}^{T}(t) P \widehat{x}(t)+\frac{1}{2} u^{T}(t) Q u(t)\right] d t+\frac{1}{2} \widehat{x}^{T}(T) X_{T} \widehat{x}(T)\right]+\alpha
其中 alpha\alpha 是一个与控制器无关的常数。
这个等效问题可以使用经典随机控制理论中的标准方法来解决 [93]。最优控制由以下公式给出:
u^(***)(t)=K_(t)^(***)(y_(0)(s),0 <= s <= t)=-Q^(-1)B_(2)^(T)X(t) widehat(x)(t)u^{\star}(t)=K_{t}^{\star}\left(y_{0}(s), 0 \leq s \leq t\right)=-Q^{-1} B_{2}^{T} X(t) \widehat{x}(t)
P[int_(0)^(T)(beta_(z)^(T)(t)beta_(z)(t)+epsiloneta^(T)(t)eta(t))dt] <= (g^(2)-epsilon)P[int_(0)^(T)beta_(w)^(T)(t)beta_(w)(t)dt]+mu_(1)+mu_(2)t\mathbb{P}\left[\int_{0}^{T}\left(\beta_{z}^{T}(t) \beta_{z}(t)+\epsilon \eta^{T}(t) \eta(t)\right) d t\right] \leq\left(g^{2}-\epsilon\right) \mathbb{P}\left[\int_{0}^{T} \beta_{w}^{T}(t) \beta_{w}(t) d t\right]+\mu_{1}+\mu_{2} t
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