量子退相干性

来自集智百科 - 复杂系统|人工智能|复杂科学|复杂网络|自组织
跳到导航 跳到搜索

此词条暂由水流心不竞初译,翻译字数共,未经审校,带来阅读不便,请见谅。

模板:简述 模板:Use British English 模板:使用英式英语 模板:使用dmy日期

文件:DecoherenceQuantumClassical en.svg
In classical scattering of a target body by environmental photons, the motion of the target body will not be changed by the scattered photons on the average. In quantum scattering, the interaction between the scattered photons and the superposed target body will cause them to be entangled, thereby delocalizing the phase coherence from the target body to the whole system, rendering the interference pattern unobservable.

thumb | 200px |在环境光子对目标物体的经典散射中,散射光子对目标物体的运动平均不产生影响。在量子散射中,散射光子与叠加靶体的相互作用会使它们纠缠在一起,从而使从靶体到整个系统的相位相干性离域,使干涉图样不可观测。

In classical scattering of a target body by environmental photons, the motion of the target body will not be changed by the scattered photons on the average. In quantum scattering, the interaction between the scattered photons and the superposed target body will cause them to be entangled, thereby delocalizing the phase coherence from the target body to the whole system, rendering the interference pattern unobservable.

在环境光子对靶体的经典散射中,靶体的运动平均不受散射光子的影响。在量子散射中,散射光子与叠加的靶体之间的相互作用会使它们产生纠缠,从而使相位相干从靶体向整个系统离域,使干涉图样变得不可观测。

模板:Quantum mechanics 模板:量子力学

Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

“量子退相干”是指量子相干的损失。在量子力学中,粒子电子波函数来描述,波函数是系统量子态的数学表示;波函数的概率解释用于解释各种量子效应。只要不同态之间存在一定的相位关系,系统就称为相干系统。在量子态编码的量子信息上进行量子计算需要一个确定的相位关系。相干性在量子物理定律下保持不变。

Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

量子退相干是量子相干性的损失。在量子力学中,粒子如电子被描述为波函数,波函数是系统量子态的数学表示; 波函数的概率解释被用来解释各种量子效应。只要不同态之间存在一定的相位关系,系统就是相干的。量子信息在量子态中的编码需要一个确定的相位关系。相干性在量子物理定律下是保持不变的。


If a quantum system were perfectly isolated, it would maintain coherence indefinitely, but it would be impossible to manipulate or investigate it. If it is not perfectly isolated, for example during a measurement, coherence is shared with the environment and appears to be lost with time; a process called quantum decoherence. As a result of this process, quantum behavior is apparently lost, just as energy appears to be lost by friction in classical mechanics.

If a quantum system were perfectly isolated, it would maintain coherence indefinitely, but it would be impossible to manipulate or investigate it. If it is not perfectly isolated, for example during a measurement, coherence is shared with the environment and appears to be lost with time; a process called quantum decoherence. As a result of this process, quantum behavior is apparently lost, just as energy appears to be lost by friction in classical mechanics.

如果一个量子系统是完全孤立的,它将无限期地保持相干性,但它将不可能操纵或调查它。如果它不是完全孤立的,例如在测量过程中,连贯性是与环境共享的,并且似乎随着时间而丢失; 这个过程被称为量子退相干。作为这个过程的结果,量子行为明显地丢失了,就像能量似乎在经典力学的摩擦中丢失了一样。


Decoherence was first introduced in 1970 by the German physicist H. Dieter Zeh[1] and has been a subject of active research since the 1980s.[2] Decoherence has been developed into a complete framework, but it does not solve the measurement problem, as the founders of decoherence theory admit in their seminal papers.[3]

对退相干的回顾,Joos(1999)指出“退相干解决了测量问题吗?显然不是。退相干告诉我们,当观察到某些物体时,它们看起来是经典的。但什么是观察?在某个阶段,我们仍然需要应用量子理论的概率法则。Adler, Stephen L. (2003). "Why decoherence has not solved the measurement problem: a response to P.W. Anderson". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 34 (1): 135–142. arXiv:quant-ph/0112095. Bibcode:2003SHPMP..34..135A. doi:10.1016/S1355-2198(02)00086-2. Unknown parameter |s2cid= ignored (help)</ref>

Decoherence can be viewed as the loss of information from a system into the environment (often modeled as a heat bath), since every system is loosely coupled with the energetic state of its surroundings. Viewed in isolation, the system's dynamics are non-unitary (although the combined system plus environment evolves in a unitary fashion). Thus the dynamics of the system alone are irreversible. As with any coupling, entanglements are generated between the system and environment. These have the effect of sharing quantum information with—or transferring it to—the surroundings.

退相干可以被看作是一个系统向环境中的信息丢失(通常被模拟为一个热浴池) ,因为每个系统都与其周围环境的能量状态松散耦合。孤立地看,系统的动态是非一元的(尽管组合系统加上环境以一元的方式进化)。因此,系统的动力学本身是不可逆的。与任何耦合一样,在系统和环境之间产生纠缠。它们具有与周围环境共享或传递量子信息的作用。

Decoherence can be viewed as the loss of information from a system into the environment (often modeled as a heat bath),[4] since every system is loosely coupled with the energetic state of its surroundings. Viewed in isolation, the system's dynamics are non-unitary (although the combined system plus environment evolves in a unitary fashion).[5] Thus the dynamics of the system alone are irreversible. As with any coupling, entanglements are generated between the system and environment. These have the effect of sharing quantum information with—or transferring it to—the surroundings.

退相干可被视为从系统到环境中的信息丢失(通常建模为热浴),[4]因为每个系统都是与其周围的能量状态松散耦合的。孤立地看,系统的动力学是非-(尽管组合系统和环境以酉的方式发展)[5] 因此,系统本身的动力学是不可逆。与任何耦合一样,纠缠在系统和环境之间产生。它们具有与周围环境共享量子信息或将其转移到周围环境的效果。

Decoherence has been used to understand the collapse of the wave function in quantum mechanics. Decoherence does not generate actual wave-function collapse. It only provides an explanation for apparent wave-function collapse, as the quantum nature of the system "leaks" into the environment. That is, components of the wave function are decoupled from a coherent system and acquire phases from their immediate surroundings. A total superposition of the global or universal wavefunction still exists (and remains coherent at the global level), but its ultimate fate remains an interpretational issue. Specifically, decoherence does not attempt to explain the measurement problem. Rather, decoherence provides an explanation for the transition of the system to a mixture of states that seem to correspond to those states observers perceive. Moreover, our observation tells us that this mixture looks like a proper quantum ensemble in a measurement situation, as we observe that measurements lead to the "realization" of precisely one state in the "ensemble".

退相干被用来理解量子力学中波函数的崩塌。退相干并不产生实际的波函数崩溃,它只是提供了一个明显的波函数崩溃的解释,因为系统的量子本质“泄漏”到环境中。也就是说,波函数的组成部分与相干系统解耦,并从其周围环境中获取相位。全局波函数或普适波函数的完全叠加仍然存在(并且在全局水平上仍然是相干的) ,但它的最终命运仍然是一个解释问题。具体来说,退相干并不试图解释测量问题。相反,退相干解释了系统向混合状态的转变,这些状态似乎与观察者感知到的状态相对应。此外,我们的观察告诉我们,这种混合物在测量情况下看起来像一个正常的量子系统,因为我们观察到测量导致精确的“系统”中的一个状态的“实现”。

Decoherence has been used to understand the collapse of the wave function in quantum mechanics. Decoherence does not generate actual wave-function collapse. It only provides an explanation for apparent wave-function collapse, as the quantum nature of the system "leaks" into the environment. That is, components of the wave function are decoupled from a coherent system and acquire phases from their immediate surroundings. A total superposition of the global or universal wavefunction still exists (and remains coherent at the global level), but its ultimate fate remains an interpretational issue. Specifically, decoherence does not attempt to explain the measurement problem. Rather, decoherence provides an explanation for the transition of the system to a mixture of states that seem to correspond to those states observers perceive. Moreover, our observation tells us that this mixture looks like a proper quantum ensemble in a measurement situation, as we observe that measurements lead to the "realization" of precisely one state in the "ensemble".

退相干已经被用来理解量子力学中的波函数的崩溃。退相干不会产生“实际”波函数崩溃。它只解释了“明显的”波函数崩溃,因为系统的量子本质“泄漏”到环境中。也就是说,波函数的各分量与相干系统分离,并从其周围环境获得相位。整体或普遍波函数的完全叠加仍然存在(并且在整体水平上保持一致),但其最终命运仍然是解释问题。具体来说,退相干并不试图解释测量问题。相反,退相干解释了系统向混合态的转变,这种转变似乎与观察者所感知的状态相对应。此外,我们的观察告诉我们,这种混合物在测量情况下看起来像一个合适的量子系综,因为我们观察到测量导致“系综”中恰好一个状态的“实现”。

Decoherence represents a challenge for the practical realization of quantum computers, since such machines are expected to rely heavily on the undisturbed evolution of quantum coherences. Simply put, they require that the coherence of states be preserved and that decoherence is managed, in order to actually perform quantum computation. The preservation of coherence, and mitigation of decoherence effects, are thus related to the concept of quantum error correction.

退相干对于量子计算机的实际实现来说是一个挑战,因为人们期望这种机器严重依赖于量子相干不受干扰的进化。简单地说,他们要求保持状态的相干性,并管理退相干,以便实际执行量子计算。因此,保持相干性和减少退相干效应与量子纠错的概念有关。

Decoherence represents a challenge for the practical realization of quantum computers, since such machines are expected to rely heavily on the undisturbed evolution of quantum coherences. Simply put, they require that the coherence of states be preserved and that decoherence is managed, in order to actually perform quantum computation. The preservation of coherence, and mitigation of decoherence effects, are thus related to the concept of quantum error correction.

退相干对实际实现量子计算机是一个挑战,因为这类机器在很大程度上依赖于量子相干的无干扰演化。简单地说,它们要求保持态的相干性,并管理退相干,以便实际执行量子计算。因此,相干的保持和退相干效应的缓解与量子纠错的概念有关。

Mechanisms机制

To examine how decoherence operates, an "intuitive" model is presented. The model requires some familiarity with quantum theory basics. Analogies are made between visualisable classical phase spaces and Hilbert spaces. A more rigorous derivation in Dirac notation shows how decoherence destroys interference effects and the "quantum nature" of systems. Next, the density matrix approach is presented for perspective.

为了检验退相干是如何运作的,提出一个“直观的”模型。这个模型需要一些量子理论的基础知识。在可视化的经典相空间和希尔伯特空间之间进行类比。狄拉克符号中一个更严格的推导显示了退相干如何破坏干涉效应和系统的“量子本质”。其次,提出了透视的密度矩阵方法。

To examine how decoherence operates, an "intuitive" model is presented. The model requires some familiarity with quantum theory basics. Analogies are made between visualisable classical phase spaces and Hilbert spaces. A more rigorous derivation in Dirac notation shows how decoherence destroys interference effects and the "quantum nature" of systems. Next, the density matrix approach is presented for perspective.

为了检验退相干是如何工作的,我们提出了一个“直观”的模型。这个模型需要熟悉一些量子理论的基础知识。在可视的经典相空间s和希尔伯特空间s之间进行了类比。在狄拉克符号中更严格的推导说明了退相干如何破坏干扰效应和系统的“量子性质”。接下来,给出了透视图的密度矩阵方法。

Rabi oscillations]]

拉比振荡]]

文件:Quantum superposition of states and decoherence.ogv
Quantum superposition of states and decoherence measurement through Rabi oscillations

[[文件:量子叠加国家和退相干.ogv|thumb |直立=1.5 |态的量子叠加和通过拉比振荡]的退相干测量。

Phase-space picture相空间图

An N-particle system can be represented in non-relativistic quantum mechanics by a wave function [math]\displaystyle{ \psi(x_1, x_2, \dots, x_N) }[/math], where each xi is a point in 3-dimensional space. This has analogies with the classical phase space. A classical phase space contains a real-valued function in 6N dimensions (each particle contributes 3 spatial coordinates and 3 momenta). Our "quantum" phase space, on the other hand, involves a complex-valued function on a 3N-dimensional space. The position and momenta are represented by operators that do not commute, and [math]\displaystyle{ \psi }[/math] lives in the mathematical structure of a Hilbert space. Aside from these differences, however, the rough analogy holds.

一个 N 粒子系统可以用一个波函数 [math]\displaystyle{ \psi(x_1, x_2, \dots, x_N) }[/math] 来表示,其中每个 xi 是三维空间中的一个点。这与经典的相空间有相似之处。一个经典的相空间包含一个6N 维的实值函数(每个粒子分配3个空间坐标和3个动量)。另一方面,我们的“量子”相空间涉及到一个3N 维空间上的复值函数。位置和动量由不交换的操作符表示,并且存在于希尔伯特空间的数学结构中。然而,撇开这些差异不谈,这个粗略的类比还是成立的。

An N-particle system can be represented in non-relativistic quantum mechanics by a wave function [math]\displaystyle{ \psi(x_1, x_2, \dots, x_N) }[/math], where each xi is a point in 3-dimensional space. This has analogies with the classical phase space. A classical phase space contains a real-valued function in 6N dimensions (each particle contributes 3 spatial coordinates and 3 momenta). Our "quantum" phase space, on the other hand, involves a complex-valued function on a 3N-dimensional space. The position and momenta are represented by operators that do not commute, and [math]\displaystyle{ \psi }[/math] lives in the mathematical structure of a Hilbert space. Aside from these differences, however, the rough analogy holds.

在非相对论量子力学中,“N”粒子系统可以用波函数[math]\displaystyle{ \psi(x\u 1,x\u 2,\dots,x\u N) }[/math]来表示,其中每个“xi”是三维空间中的一个点。这与经典的相空间类似。一个经典的相空间包含一个6N维的实值函数(每个粒子贡献3个空间坐标和3个动量)。另一方面,我们的“量子”相空间涉及一个3N维空间上的复值函数。位置和动量由不交换的算子表示,[math]\displaystyle{ \psi }[/math]存在于希尔伯特空间的数学结构中。然而,除了这些差异之外,粗略的类比是成立的。

Different previously isolated, non-interacting systems occupy different phase spaces. Alternatively we can say that they occupy different lower-dimensional subspaces in the phase space of the joint system. The effective dimensionality of a system's phase space is the number of degrees of freedom present, which—in non-relativistic models—is 6 times the number of a system's free particles. For a macroscopic system this will be a very large dimensionality. When two systems (and the environment would be a system) start to interact, though, their associated state vectors are no longer constrained to the subspaces. Instead the combined state vector time-evolves a path through the "larger volume", whose dimensionality is the sum of the dimensions of the two subspaces. The extent to which two vectors interfere with each other is a measure of how "close" they are to each other (formally, their overlap or Hilbert space multiplies together) in the phase space. When a system couples to an external environment, the dimensionality of, and hence "volume" available to, the joint state vector increases enormously. Each environmental degree of freedom contributes an extra dimension.

不同的先前孤立的,非相互作用系统占据不同的相空间。或者我们可以说,它们在联合系统的相空间中占据不同的低维子空间。系统相空间的有效维数是自由度数,在非相对论模型中,自由度数是系统自由粒子数的6倍。对于宏观系统,这是一个非常大的维数。但是,当两个系统(环境是一个系统)开始交互时,它们相关的状态向量不再局限于子空间。相反,合并的状态向量时间演化一条通过“更大的体积”的路径,其维数是两个子空间维数之和。两个矢量相互干涉的程度是衡量它们在相空间中相互距离(形式上,它们的重叠或希尔伯特空间相乘)的一个尺度。当一个系统与外部环境耦合时,联合状态向量的维数以及可用的“体积”都会大大增加。每一个环境自由度都增加了一个额外的维度。

Different previously isolated, non-interacting systems occupy different phase spaces. Alternatively we can say that they occupy different lower-dimensional subspaces in the phase space of the joint system. The effective dimensionality of a system's phase space is the number of degrees of freedom present, which—in non-relativistic models—is 6 times the number of a system's free particles. For a macroscopic system this will be a very large dimensionality. When two systems (and the environment would be a system) start to interact, though, their associated state vectors are no longer constrained to the subspaces. Instead the combined state vector time-evolves a path through the "larger volume", whose dimensionality is the sum of the dimensions of the two subspaces. The extent to which two vectors interfere with each other is a measure of how "close" they are to each other (formally, their overlap or Hilbert space multiplies together) in the phase space. When a system couples to an external environment, the dimensionality of, and hence "volume" available to, the joint state vector increases enormously. Each environmental degree of freedom contributes an extra dimension.


The original system's wave function can be expanded in many different ways as a sum of elements in a quantum superposition. Each expansion corresponds to a projection of the wave vector onto a basis. The basis can be chosen at will. Let us choose an expansion where the resulting basis elements interact with the environment in an element-specific way. Such elements will—with overwhelming probability—be rapidly separated from each other by their natural unitary time evolution along their own independent paths. After a very short interaction, there is almost no chance of any further interference. The process is effectively irreversible. The different elements effectively become "lost" from each other in the expanded phase space created by coupling with the environment; in phase space, this decoupling is monitored through the Wigner quasi-probability distribution. The original elements are said to have decohered. The environment has effectively selected out those expansions or decompositions of the original state vector that decohere (or lose phase coherence) with each other. This is called "environmentally-induced superselection", or einselection. The decohered elements of the system no longer exhibit quantum interference between each other, as in a double-slit experiment. Any elements that decohere from each other via environmental interactions are said to be quantum-entangled with the environment. The converse is not true: not all entangled states are decohered from each other.

原始系统的波函数可以以许多不同的方式展开,作为一个态叠加原理的元素之和。每个展开式对应于波矢到基的一个投影。基础可以随意选择。让我们选择一个扩展,其中产生的基本元素以特定于元素的方式与环境交互。这些元素将以压倒性的概率,通过它们自身独立的路径上的自然幺正时间演化而迅速地彼此分离。在非常短的相互作用之后,几乎没有任何进一步干扰的可能。这个过程实际上是不可逆的。在与环境耦合产生的扩展相空间中,不同的元素有效地相互“丢失” ,在相空间中,通过维格纳准概率分布对这种解耦进行监测。据说最初的元素已经被解密了。环境有效地选出了原始状态向量相互解码(或失去相位一致性)的展开或分解。这被称为“环境诱导的超级选择” ,或单一选择。系统中的退相干元件之间不再像双缝实验那样存在量子干涉。任何通过环境相互作用彼此解码的元素都被称为与环境的量子纠缠。反之则不然: 并非所有的纠缠态都彼此退相干。

The original system's wave function can be expanded in many different ways as a sum of elements in a quantum superposition. Each expansion corresponds to a projection of the wave vector onto a basis. The basis can be chosen at will. Let us choose an expansion where the resulting basis elements interact with the environment in an element-specific way. Such elements will—with overwhelming probability—be rapidly separated from each other by their natural unitary time evolution along their own independent paths. After a very short interaction, there is almost no chance of any further interference. The process is effectively irreversible. The different elements effectively become "lost" from each other in the expanded phase space created by coupling with the environment; in phase space, this decoupling is monitored through the Wigner quasi-probability distribution. The original elements are said to have decohered. The environment has effectively selected out those expansions or decompositions of the original state vector that decohere (or lose phase coherence) with each other. This is called "environmentally-induced superselection", or einselection.[6] The decohered elements of the system no longer exhibit quantum interference between each other, as in a double-slit experiment. Any elements that decohere from each other via environmental interactions are said to be quantum-entangled with the environment. The converse is not true: not all entangled states are decohered from each other.


Any measuring device or apparatus acts as an environment, since at some stage along the measuring chain, it has to be large enough to be read by humans. It must possess a very large number of hidden degrees of freedom. In effect, the interactions may be considered to be quantum measurements. As a result of an interaction, the wave functions of the system and the measuring device become entangled with each other. Decoherence happens when different portions of the system's wave function become entangled in different ways with the measuring device. For two einselected elements of the entangled system's state to interfere, both the original system and the measuring in both elements device must significantly overlap, in the scalar product sense. If the measuring device has many degrees of freedom, it is very unlikely for this to happen.

任何测量设备或仪器都充当环境的角色,因为在测量链的某个阶段,它必须足够大以便人类能够读取。它必须具有大量的隐藏自由度。实际上,相互作用可以认为是量子测量。由于相互作用,系统和测量装置的波函数相互纠缠。当系统波函数的不同部分以不同的方式与测量装置发生纠缠时,就会发生消相干。对于纠缠系统中的两个单选元素进行干涉,在标量积意义上,原始系统和两个元素中的测量设备都必须有显著的重叠。如果测量装置有许多自由度,这是不太可能发生的。

Any measuring device or apparatus acts as an environment, since at some stage along the measuring chain, it has to be large enough to be read by humans. It must possess a very large number of hidden degrees of freedom. In effect, the interactions may be considered to be quantum measurements. As a result of an interaction, the wave functions of the system and the measuring device become entangled with each other. Decoherence happens when different portions of the system's wave function become entangled in different ways with the measuring device. For two einselected elements of the entangled system's state to interfere, both the original system and the measuring in both elements device must significantly overlap, in the scalar product sense. If the measuring device has many degrees of freedom, it is very unlikely for this to happen.


As a consequence, the system behaves as a classical statistical ensemble of the different elements rather than as a single coherent quantum superposition of them. From the perspective of each ensemble member's measuring device, the system appears to have irreversibly collapsed onto a state with a precise value for the measured attributes, relative to that element. And this, provided one explains how the Born rule coefficients effectively act as probabilities as per the measurement postulate, constitutes a solution to the quantum measurement problem.

因此,该系统表现为不同元素的经典系综,而不是它们的单一连贯态叠加原理。从每个成员的测量设备的角度来看,系统似乎已经不可逆转地崩溃到一个精确值的测量属性的状态,相对于该元素。如果能解释波恩规则系数如何根据测量假设有效地作为概率,这就构成了量子测量问题的解决方案。

As a consequence, the system behaves as a classical statistical ensemble of the different elements rather than as a single coherent quantum superposition of them. From the perspective of each ensemble member's measuring device, the system appears to have irreversibly collapsed onto a state with a precise value for the measured attributes, relative to that element. And this, provided one explains how the Born rule coefficients effectively act as probabilities as per the measurement postulate, constitutes a solution to the quantum measurement problem.


Dirac notation狄拉克符号

Using Dirac notation, let the system initially be in the state

使用狄拉克符号,让系统最初处于这种状态

Using Dirac notation, let the system initially be in the state


[math]\displaystyle{ |\psi\rang = \sum_i |i\rang \lang i |\psi\rang, }[/math]

[数学][数学]

[math]\displaystyle{ |\psi\rang = \sum_i |i\rang \lang i |\psi\rang, }[/math]


where the [math]\displaystyle{ |i\rang }[/math]s form an einselected basis (environmentally induced selected eigenbasis

在那里的“数学” | 我呼叫“数学”形成一个单一的选择基础(环境诱导选择特征基础

where the [math]\displaystyle{ |i\rang }[/math]s form an einselected basis (environmentally induced selected eigenbasis[6]), and let the environment initially be in the state [math]\displaystyle{ |\epsilon\rang }[/math]. The vector basis of the combination of the system and the environment consists of the tensor products of the basis vectors of the two subsystems. Thus, before any interaction between the two subsystems, the joint state can be written as


In terms of density matrices, the loss of interference effects corresponds to the diagonalization of the "environmentally traced-over" density matrix.

就密度矩阵而言,干扰效应的损失相当于“环境追踪”密度矩阵的对角化。

[math]\displaystyle{ |\text{before}\rang = \sum_i |i\rang |\epsilon\rang \lang i|\psi\rang, }[/math]


[math]\displaystyle{ \rho_\text{sys} = \operatorname{Tr}_\text{env}\Big(\sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \otimes |\epsilon_i\rang \lang\epsilon_j|\Big) = \sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \lang\epsilon_j|\epsilon_i\rang  = \sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \delta_{ij} = \sum_i |\psi_i|^2 |i\rang \lang i|. }[/math]

比格(sum { i,j } psi i psi ^ j ^ * * * | i rang j | otimes | epsilon i rang lang epsilon | j | Big) = sum { i,j } psi psi psi psi psi ^ * * ^ | 我给 lang 打了电话我们会在这里等你,我们会在这里我打电话给萨姆,希望他能回来我给 lang 打了电话三角洲{ ij } = sum i | 我不知道你会怎么做回头见我给朗格打了电话。数学

where [math]\displaystyle{ |i\rang |\epsilon\rang }[/math] is shorthand for the tensor product [math]\displaystyle{ |i\rang \otimes |\epsilon\rang }[/math]. There are two extremes in the way the system can interact with its environment: either (1) the system loses its distinct identity and merges with the environment (e.g. photons in a cold, dark cavity get converted into molecular excitations within the cavity walls), or (2) the system is not disturbed at all, even though the environment is disturbed (e.g. the idealized non-disturbing measurement). In general, an interaction is a mixture of these two extremes that we examine.


Similarly, the final reduced density matrix after the transition will be

同样,最终降低密度矩阵后的转变也会

System absorbed by environment环境吸收系统

If the environment absorbs the system, each element of the total system's basis interacts with the environment such that

如果环境吸收了系统,那么整个系统基础的每个元素都与环境相互作用,从而

[math]\displaystyle{ \sum_j |\phi_j|^2 |j\rang \lang j|. }[/math]

[数学,数学]


[math]\displaystyle{ |i\rang |\epsilon\rang }[/math] evolves into [math]\displaystyle{ |\epsilon_i\rang, }[/math]

The transition probability will then be given as

然后给出转移概率


and so

于是

[math]\displaystyle{ \operatorname{prob}_\text{after}(\psi \to \phi) = \sum_{i,j} |\psi_i|^2 |\phi_j|^2 \lang j|i\rang \lang i|j\rang = \sum_i |\psi_i^* \phi_i|^2, }[/math]



[math]\displaystyle{ |\text{before}\rang }[/math] evolves into [math]\displaystyle{ |\text{after}\rang = \sum_i |\epsilon_i\rang \lang i|\psi\rang. }[/math]

which has no contribution from the interference terms

干涉条款没有任何贡献


The unitarity of time evolution demands that the total state basis remains orthonormal, i.e. the scalar or inner products of the basis vectors must vanish, since

时间演化的unitarity要求总状态基保持orthonormal,即基向量的 scalarinner products必须消失,因为

[math]\displaystyle{ \lang i|j\rang = \delta_{ij} }[/math]:

[math]\displaystyle{ \sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i. }[/math]


[math]\displaystyle{ \lang\epsilon_i|\epsilon_j\rang = \delta_{ij}. }[/math]

The density-matrix approach has been combined with the Bohmian approach to yield a reduced-trajectory approach, taking into account the system reduced density matrix and the influence of the environment.

密度矩阵方法与 Bohmian 方法相结合,得到了一种简化轨迹方法,同时考虑了系统简化密度矩阵和环境的影响。


This orthonormality of the environment states is the defining characteristic required for einselection.[6]

环境状态的这种正交性是einselection所需的定义特征。[6]

Consider a system S and environment (bath) B, which are closed and can be treated quantum-mechanically. Let [math]\displaystyle{ \mathcal H_S }[/math] and [math]\displaystyle{ \mathcal H_B }[/math] be the system's and bath's Hilbert spaces respectively. Then the Hamiltonian for the combined system is

考虑一个系统 s 和环境(浴室) b,它们是封闭的,可以用量子力学方法处理。设系统的希尔伯特空间为数学空间,巴斯的希尔伯特空间为数学空间,数学空间为数学空间。那么组合系统的哈密顿量是

System not disturbed by environment系统不受环境干扰

In an idealised measurement, the system disturbs the environment, but is itself undisturbed by the environment.

在理想化的测量中,系统会干扰环境,但本身不会受到环境的干扰。

[math]\displaystyle{ \hat{H} = \hat H_S \otimes \hat I_B + \hat I_S \otimes \hat H_B + \hat H_I, }[/math]

有时候我觉得自己是个好人,有时候我觉得自己是个好人

In this case, each element of the basis interacts with the environment such that

在这种情况下,基的每个元素都与环境交互,从而

where [math]\displaystyle{ \hat H_S, \hat H_B }[/math] are the system and bath Hamiltonians respectively, [math]\displaystyle{ \hat H_I }[/math] is the interaction Hamiltonian between the system and bath, and [math]\displaystyle{ \hat I_S, \hat I_B }[/math] are the identity operators on the system and bath Hilbert spaces respectively. The time-evolution of the density operator of this closed system is unitary and, as such, is given by

其中 h _ s,h _ b 分别是系统和沐浴哈密顿函数,h _ i 分别是系统和沐浴哈密顿函数之间的相互作用哈密顿函数,而 h _ s,h _ b 分别是系统和沐浴希尔伯特空间上的恒等算子。这个封闭系统的密度算符的时间演化是幺正的,因此,给出了

[math]\displaystyle{ |i\rang |\epsilon\rang }[/math] evolves into the product [math]\displaystyle{ |i, \epsilon_i\rang = |i\rang |\epsilon_i\rang, }[/math]


[math]\displaystyle{ \rho_{SB}(t) = \hat U(t) \rho_{SB}(0) \hat U^\dagger(t), }[/math]

< math > rho { SB }(t) = hat u (t) rho { SB }(0) hat u ^ dagger (t) ,</math >

and so 于是


where the unitary operator is [math]\displaystyle{ \hat U = e^{-i\hat{H}t/\hbar} }[/math]. If the system and bath are not entangled initially, then we can write [math]\displaystyle{ \rho_{SB} = \rho_S \otimes \rho_B }[/math]. Therefore, the evolution of the system becomes

其中幺正算符是 < math > hat u = e ^ { i hat { h } t/hbar } </math > 。如果系统和浴室最初没有纠缠,那么我们可以写出 < math > rho _ (SB) = rho _ s 和 rho _ b </math > 。因此,系统的演化成为

[math]\displaystyle{ |\text{before}\rang }[/math] evolves into [math]\displaystyle{ |\text{after}\rang = \sum_i |i, \epsilon_i\rang \lang i|\psi\rang. }[/math]


[math]\displaystyle{ \rho_{SB}(t) = \hat U (t)[\rho_S(0) \otimes \rho_B(0)] \hat U^\dagger(t). }[/math]

[ rho _ { SB }(t) = hat u (t)[ rho _ s (0) o _ ho _ b (0)] hat u ^ dagger (t)

In this case, unitarity demands that

在这种情况下, unitarity要求

The system–bath interaction Hamiltonian can be written in a general form as

系统-浴相互作用哈密顿量可以写成一般形式如下:

[math]\displaystyle{ \lang i, \epsilon_i|j, \epsilon_j\rang = \lang i|j\rang \lang\epsilon_i|\epsilon_j\rang = \delta_{ij} \lang\epsilon_i|\epsilon_j\rang = \delta_{ij} \lang\epsilon_i|\epsilon_i\rang = \delta_{ij}, }[/math]


[math]\displaystyle{ \hat H_I = \sum_i \hat S_i \otimes \hat B_i, }[/math]

数学,数学,数学,数学

where [math]\displaystyle{ \lang \epsilon_i | \epsilon_i \rang = 1 }[/math] was used. Additionally, decoherence requires, by virtue of the large number of hidden degrees of freedom in the environment, that

其中使用了[math]\displaystyle{ \lang\epsilon|u i | \epsilon_i\rang=1 }[/math]此外,由于环境中隐藏了大量的自由度,退相干要求

where [math]\displaystyle{ \hat S_i \otimes \hat B_i }[/math] is the operator acting on the combined system–bath Hilbert space, and [math]\displaystyle{ \hat S_i, \hat B_i }[/math] are the operators that act on the system and bath respectively. This coupling of the system and bath is the cause of decoherence in the system alone. To see this, a partial trace is performed over the bath to give a description of the system alone:

其中 b _ i </math > 是作用于组合系统-bath Hilbert 空间的算子,而 s _ i,b _ i </math > 是分别作用于系统和 bath 的算子。系统和熔池的这种耦合是系统本身退相干的原因。为了看到这一点,在浴缸上执行部分跟踪来描述系统本身:

[math]\displaystyle{ \lang\epsilon_i|\epsilon_j\rang \approx \delta_{ij}. }[/math]


[math]\displaystyle{ \rho_S(t) = \operatorname{Tr}_B\big[\hat U(t)[\rho_S(0) \otimes \rho_B(0)] \hat U^\dagger(t)\big]. }[/math]

“ rho _ s (t) = 操作员名称{ Tr } _ b big [ hat u (t)[ rho _ s (0)或 rho _ b (0)] hat u ^ (t) big ]

As before, this is the defining characteristic for decoherence to become einselection.[6] The approximation becomes more exact as the number of environmental degrees of freedom affected increases.

与之前一样,这是退相干成为einselection的定义特征。[7]随着受影响的环境自由度的增加,近似值变得更加精确。

[math]\displaystyle{ \rho_S(t) }[/math] is called the reduced density matrix and gives information about the system only. If the bath is written in terms of its set of orthogonal basis kets, that is, if it has been initially diagonalized, then [math]\displaystyle{ \textstyle\rho_B(0) = \sum_j a_j |j\rangle \langle j| }[/math]. Computing the partial trace with respect to this (computational) basis gives

Rho _ s (t) </math > 被称为约化密度矩阵,它只给出系统的信息。如果浴缸是按照它的正交基来写的,也就是说,如果它最初是对角化的,那么 < math > textstyle rho _ b (0) = sum _ j a j | j rangle langle j | </math > 。计算关于这个(计算)基的部分跟踪给出

Note that if the system basis [math]\displaystyle{ |i\rang }[/math] were not an einselected basis, then the last condition is trivial, since the disturbed environment is not a function of [math]\displaystyle{ i }[/math], and we have the trivial disturbed environment basis [math]\displaystyle{ |\epsilon_j\rang = |\epsilon'\rang }[/math]. This would correspond to the system basis being degenerate with respect to the environmentally defined measurement observable. For a complex environmental interaction (which would be expected for a typical macroscale interaction) a non-einselected basis would be hard to define.

请注意,如果系统基础[math]\displaystyle{ |i\rang }[/math]不是一个选定的基础,那么最后一个条件是平凡的,因为受干扰的环境不是[math]\displaystyle{ i }[/math]的函数,我们有平凡的受干扰环境基础[math]\displaystyle{ | \epsilon|u j\rang=| \epsilon'\rang }[/math]。这将对应于系统基础相对于环境定义的可观测测量退化。对于一个复杂的环境相互作用(对于一个典型的宏观相互作用来说,这是预期的),一个非选择的基础将很难定义。

[math]\displaystyle{ \rho_S(t) = \sum_l \hat A_l \rho_S(0) \hat A^\dagger_l, }[/math]

[数学] ρ _ s (t) = sum _ l ρ _ s (0) hat a ^ daggl,[数学]

Loss of interference and the transition from quantum to classical probabilities干涉损失与从量子概率到经典概率的转变

The utility of decoherence lies in its application to the analysis of probabilities, before and after environmental interaction, and in particular to the vanishing of quantum interference terms after decoherence has occurred. If we ask what is the probability of observing the system making a transition from [math]\displaystyle{ \psi }[/math] to [math]\displaystyle{ \phi }[/math] before [math]\displaystyle{ \psi }[/math] has interacted with its environment, then application of the Born probability rule states that the transition probability is the squared modulus of the scalar product of the two states:

退相干的效用在于其应用于分析环境相互作用前后的概率,特别是退相干发生后量子干涉项的消失。如果我们问,在系统与其环境相互作用之前,观察系统从[math]\displaystyle{ \psi }[/math][math]\displaystyle{ \phi }[/math]跃迁的概率是多少,然后应用玻恩概率规则说明跃迁概率是两个状态的标量积的平方模:

where [math]\displaystyle{ \hat A_l, \hat A^\dagger_l }[/math] are defined as the Kraus operators and are represented as

其中,a _ l,a ^ dagger _ l 被定义为 Kraus 操作符,并表示为


[math]\displaystyle{ \operatorname{prob}_\text{before}(\psi \to \phi) = \left|\lang\psi|\phi\rang\right|^2 = \left|\sum_i \psi^*_i \phi_i\right|^2 = \sum_i |\psi_i^* \phi_i|^2 + \sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i , }[/math]
[math]\displaystyle{ \hat A_l = \sqrt{a_j} \langle k| \hat U |j\rangle. }[/math]

帽子 a l = sqrt { a j } langle k | hat u | j rangle


where [math]\displaystyle{ \psi_i = \lang i|\psi\rang }[/math], [math]\displaystyle{ \psi_i^* = \lang\psi|i\rang }[/math], and [math]\displaystyle{ \phi_i = \lang i|\phi\rang }[/math] etc.

其中[math]\displaystyle{ \psi_i = \lang i|\psi\rang }[/math], [math]\displaystyle{ \psi_i^* = \lang\psi|i\rang }[/math], and [math]\displaystyle{ \phi_i = \lang i|\phi\rang }[/math]等等。

This is known as the operator-sum representation (OSR). A condition on the Kraus operators can be obtained by using the fact that [math]\displaystyle{ \operatorname{Tr}[\rho_S(t)] = 1 }[/math]; this then gives

这就是所谓的运算符和表示(OSR)。通过使用 < math > 操作者名{ Tr }[ rho _ s (t)] = 1 </math > ,可以得到 Kraus 算子的一个条件


The above expansion of the transition probability has terms that involve [math]\displaystyle{ i \ne j }[/math]; these can be thought of as representing interference between the different basis elements or quantum alternatives. This is a purely quantum effect and represents the non-additivity of the probabilities of quantum alternatives.

上述跃迁概率的展开式包含了涉及j的项;这些项可以被认为表示不同基元或量子替代物之间的“干扰”。这是一个纯粹的量子效应,代表了量子替代概率的不可加性。

[math]\displaystyle{ \sum_l \hat A^\dagger_l \hat A_l = \hat I_S. }[/math]

[数学][数学][数学]


To calculate the probability of observing the system making a quantum leap from [math]\displaystyle{ \psi }[/math] to [math]\displaystyle{ \phi }[/math] after [math]\displaystyle{ \psi }[/math] has interacted with its environment, then application of the Born probability rule states that we must sum over all the relevant possible states [math]\displaystyle{ |\epsilon_i\rang }[/math] of the environment before squaring the modulus:

为了计算观测系统在与环境发生相互作用后从[math]\displaystyle{ \psi }[/math]跃迁到[math]\displaystyle{ \phi }[/math]的概率,然后应用Born probability规则说明,我们必须求和环境的所有相关可能状态[math]\displaystyle{ |\epsilon|i\rang }[/math],然后对模数进行平方运算:

This restriction determines whether decoherence will occur or not in the OSR. In particular, when there is more than one term present in the sum for [math]\displaystyle{ \rho_S(t) }[/math], then the dynamics of the system will be non-unitary, and hence decoherence will take place.

这个限制决定了 OSR 中是否会发生退相干。特别是,当 < math > rho _ s (t) </math > 的和中出现多个项时,系统的动力学将是非幺正的,因此退相干将发生。


[math]\displaystyle{ \operatorname{prob}_\text{after}(\psi \to \phi) = \sum_j \,\left|\lang\text{after}\right| \phi, \epsilon_j \rang|^2 = \sum_j \,\left|\sum_i \psi_i^* \lang i, \epsilon_i|\phi, \epsilon_j\rang\right|^2 = \sum_j\left|\sum_i \psi_i^* \phi_i \lang\epsilon_i|\epsilon_j\rang \right|^2. }[/math]


A more general consideration for the existence of decoherence in a quantum system is given by the master equation, which determines how the density matrix of the system alone evolves in time (see also the Belavkin equation for the evolution under continuous measurement). This uses the Schrödinger picture, where evolution of the state (represented by its density matrix) is considered. The master equation is

主方程给出了量子系统中退相干存在的更一般性的考虑,它决定了系统的密度矩阵如何在时间中演化(也见连续测量下演化的 Belavkin 方程)。这使用了薛定谔绘景模型,其中考虑了状态的演化(由其密度矩阵表示)。主方程是

The internal summation vanishes when we apply the decoherence/einselection condition [math]\displaystyle{ \lang\epsilon_i|\epsilon_j\rang \approx \delta_{ij} }[/math], and the formula simplifies to

当我们应用退相干/einselection条件[math]\displaystyle{ \lang\epsilon\u i |\epsilon\u j\rang\approx\delta{ij} }[/math]时,内部求和消失,公式简化为

[math]\displaystyle{ \rho'_S(t) = \frac{-i}{\hbar} \big[\tilde H_S, \rho_S(t)\big] + L_D \big[\rho_S(t)\big], }[/math]

[数学] rho’ _ s (t) = frac {-i }{ hbar } big [ tilde h _ s,rho _ s (t) big ] + l _ d big [ rho _ s (t) big ] ,</math >

[math]\displaystyle{ \operatorname{prob}_\text{after}(\psi \to \phi) \approx \sum_j |\psi_j^* \phi_j|^2 = \sum_i |\psi^*_i \phi_i|^2. }[/math]

where [math]\displaystyle{ \tilde H_S = H_S + \Delta }[/math] is the system Hamiltonian [math]\displaystyle{ H_S }[/math] along with a (possible) unitary contribution [math]\displaystyle{ \Delta }[/math] from the bath, and [math]\displaystyle{ L_D }[/math] is the Lindblad decohering term. The matrix elements [math]\displaystyle{ b_{\alpha\beta} }[/math] represent the elements of a positive semi-definite Hermitian matrix; they characterize the decohering processes and, as such, are called the noise parameters.

其中,哈密尔顿体系是哈密尔顿体系,还有浴缸里的一个(可能的)单元贡献。矩阵元素 < math > b { alpha beta } </math > 表示半正定埃尔米特矩阵的元素; 它们表示退相干过程,因此称为噪声参数。


or, equivalently, the decay of the purity

或者,相当于,纯度的衰退

If we compare this with the formula we derived before the environment introduced decoherence, we can see that the effect of decoherence has been to move the summation sign [math]\displaystyle{ \textstyle\sum_i }[/math] from inside of the modulus sign to outside. As a result, all the cross- or quantum interference-terms

如果我们将其与我们在环境引入退相干之前导出的公式进行比较,我们可以看到退相干的效果是将求和符号[math]\displaystyle{ \textstyle\sum\u i }[/math]从模符号的内部移动到外部。因此,所有交叉项或量子干涉]项

.

.


[math]\displaystyle{ \sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i }[/math]


We assume for the moment that the system in question consists of a subsystem A being studied and the "environment" [math]\displaystyle{ \epsilon }[/math], and the total Hilbert space is the tensor product of a Hilbert space [math]\displaystyle{ \mathcal H_A }[/math] describing A and a Hilbert space [math]\displaystyle{ \mathcal H_\epsilon }[/math] describing [math]\displaystyle{ \epsilon }[/math], that is,

我们假设系统由一个子系统 a 和一个“环境”组成,希尔伯特空间是希尔伯特空间的张量积,

have vanished from the transition-probability calculation. The decoherence has irreversibly converted quantum behaviour (additive probability amplitudes) to classical behaviour (additive probabilities). 从转移概率计算中消失了。退相干具有不可逆将量子行为(加性概率振幅s)转换为经典行为(加性概率)。[6][8][9]


and

In terms of density matrices, the loss of interference effects corresponds to the diagonalization of the "environmentally traced-over" density matrix.[6]

就密度矩阵而言,干扰效应的损失对应于“环境跟踪过”密度矩阵的对角化。[6]

[math]\displaystyle{ U\big(|\psi_2\rangle \otimes |\text{in}\rangle\big) }[/math]

U big (| psi _ 2 rangle o times | text { in } rangle big) </math >

Density-matrix approach 密度矩阵法

The effect of decoherence on density matrices is essentially the decay or rapid vanishing of the off-diagonal elements of the partial trace of the joint system's density matrix, i.e. the trace, with respect to any environmental basis, of the density matrix of the combined system and its environment. The decoherence irreversibly converts the "averaged" or "environmentally traced-over"[6] density matrix from a pure state to a reduced mixture; it is this that gives the appearance of wave-function collapse. Again, this is called "environmentally induced superselection", or einselection.[6] The advantage of taking the partial trace is that this procedure is indifferent to the environmental basis chosen.

退相干对密度矩阵的影响本质上是联合系统密度矩阵部分迹线非对角元素的衰减或快速消失,即迹线,关于组合系统的密度矩阵的“任何”环境基础它的环境。退相干不可逆将“平均”或“环境跟踪”[6]密度矩阵从纯状态转换为还原混合物;正是这一点导致了波函数崩溃的“外观”。同样,这被称为“环境诱导的超选择”,或einselection[6]采用部分跟踪的优点是,该过程与所选的环境基础无关。

uniquely as

独一无二的


Initially, the density matrix of the combined system can be denoted as

最初,组合系统的密度矩阵可以表示为

[math]\displaystyle{ \sum_i |e_i\rangle \otimes |f_{1i}\rangle }[/math]

数学,数学,数学


[math]\displaystyle{ \rho = |\text{before}\rang \lang\text{before}| = |\psi\rang \lang\psi| \otimes |\epsilon\rang \lang\epsilon|, }[/math]

and


where [math]\displaystyle{ |\epsilon\rang }[/math] is the state of the environment. 其中[math]\displaystyle{ |\epsilon\rang }[/math] 是环境状况。

[math]\displaystyle{ \sum_i |e_i\rangle \otimes |f_{2i}\rangle }[/math]

数学,数学,数学

Then if the transition happens before any interaction takes place between the system and the environment, the environment subsystem has no part and can be traced out, leaving the reduced density matrix for the system:

如果在系统和环境之间发生任何相互作用之前发生了跃迁,则环境子系统没有部分,可以追踪,留下系统的约化密度矩阵:

respectively. One thing to realize is that the environment contains a huge number of degrees of freedom, a good number of them interacting with each other all the time. This makes the following assumption reasonable in a handwaving way, which can be shown to be true in some simple toy models. Assume that there exists a basis for [math]\displaystyle{ \mathcal H_\epsilon }[/math] such that [math]\displaystyle{ |f_{1i}\rangle }[/math] and [math]\displaystyle{ |f_{1j}\rangle }[/math] are all approximately orthogonal to a good degree if i ≠ j and the same thing for [math]\displaystyle{ |f_{2i}\rangle }[/math] and [math]\displaystyle{ |f_{2j}\rangle }[/math] and also for [math]\displaystyle{ |f_{1i}\rangle }[/math] and [math]\displaystyle{ |f_{2j}\rangle }[/math] for any i and j (the decoherence property).

相应地。要认识到的一点是,环境中包含着大量的自由度,其中很多自由度一直在相互作用。这使得下面的假设在手工方式上是合理的,这在一些简单的玩具模型中可以被证明是正确的。假设存在[math]\displaystyle{ \mathcal H_\epsilon }[/math]使得[math]\displaystyle{ |f_{1i}\rangle }[/math][math]\displaystyle{ |f_{1j}\rangle }[/math]都是近似于完全正交,如果 i ≠ j,对于任何i和j,对于[math]\displaystyle{ |f_{2i}\rangle }[/math][math]\displaystyle{ |f_{2j}\rangle }[/math]的同样特性对于[math]\displaystyle{ | f{2i}\rangle }[/math][math]\displaystyle{ |f{2j}\rangle }[/math]也一样成立(退相干特性)。


[math]\displaystyle{ \rho_\text{sys} = \operatorname{Tr}_\textrm{env}(\rho) = |\psi\rang \lang\psi| \lang\epsilon|\epsilon\rang = |\psi\rang \lang\psi|. }[/math]


This often turns out to be true (as a reasonable conjecture) in the position basis because how A interacts with the environment would often depend critically upon the position of the objects in A. Then, if we take the partial trace over the environment, we would find the density state is approximately described by

这在位置基础上通常被证明是正确的(作为一个合理的猜想) ,因为 a 与环境的相互作用往往严格取决于 a 中对象的位置。然后,如果我们对环境进行部分跟踪,我们会发现密度状态大致描述为

Now the transition probability will be given as

现在跃迁概率将被给出为

[math]\displaystyle{ \sum_i \big(\langle f_{1i}|f_{1i}\rangle + \langle f_{2i}|f_{2i}\rangle\big) |e_i\rangle \langle e_i|, }[/math]

[math]\displaystyle{ \sum_i \big(\langle f_{1i}|f_{1i}\rangle + \langle f_{2i}|f_{2i}\rangle\big) |e_i\rangle \langle e_i|, }[/math]

[math]\displaystyle{ \operatorname{prob}_\text{before}(\psi \to \phi) = \lang\phi| \rho_\text{sys} |\phi\rang = \lang\phi|\psi\rang \lang\psi|\phi\rang = \big|\lang\psi|\phi\rang\big|^2 = \sum_i |\psi_i^* \phi_i|^2 + \sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j\phi_i, }[/math]


that is, we have a diagonal mixed state, there is no constructive or destructive interference, and the "probabilities" add up classically. The time it takes for U(t) (the unitary operator as a function of time) to display the decoherence property is called the decoherence time.

也就是说,我们有一个对角线混合状态,没有建设性或破坏性干涉,而且“概率”是经典的加法。U (t)(幺正算符随时间变化的函数)显示消相干属性所需的时间称为消相干时间。

where [math]\displaystyle{ \psi_i = \lang i|\psi\rang }[/math], [math]\displaystyle{ \psi_i^* = \lang \psi|i\rang }[/math], and [math]\displaystyle{ \phi_i = \lang i|\phi\rang }[/math] etc.

其中[math]\displaystyle{ \psi_i = \lang i|\psi\rang }[/math], [math]\displaystyle{ \psi_i^* = \lang \psi|i\rang }[/math], 和 [math]\displaystyle{ \phi_i = \lang i|\phi\rang }[/math]等等。

Now the case when transition takes place after the interaction of the system with the environment. The combined density matrix will be

现在是系统与环境交互后发生转换的情况。组合密度矩阵为

[math]\displaystyle{ \rho = |\text{after}\rang \lang\text{after}| = \sum_{i,j} \psi_i \psi_j^* |i, \epsilon_i\rang \lang j, \epsilon_j| = \sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \otimes |\epsilon_i\rang \lang\epsilon_j|. }[/math]

The decoherence rate depends on a number of factors, including temperature or uncertainty in position, and many experiments have tried to measure it depending on the external environment.

退相干速率取决于许多因素,包括温度或位置的不确定性,许多实验试图根据外部环境来测量它。


To get the reduced density matrix of the system, we trace out the environment and employ the decoherence/einselection condition and see that the off-diagonal terms vanish (a result obtained by Erich Joos and H. D. Zeh in 1985):[10]

为了得到系统的约化密度矩阵,我们追踪环境,采用退相干/einselection条件,并看到非对角项消失(Erich Joos和H.D.Zeh在1985年获得的结果):[10]

The process of a quantum superposition gradually obliterated by decoherence was quantitatively measured for the first time by Serge Haroche and his co-workers at the École Normale Supérieure in Paris in 1996. Their approach involved sending individual rubidium atoms, each in a superposition of two states, through a microwave-filled cavity. The two quantum states both cause shifts in the phase of the microwave field, but by different amounts, so that the field itself is also put into a superposition of two states. Due to photon scattering on cavity-mirror imperfection, the cavity field loses phase coherence to the environment.

1996年,Serge Haroche 和他的同事们在巴黎的态叠加原理高等师范学校首次定量地测量了退相干逐渐消失的过程。他们的方法是通过一个充满微波的空腔,将处于两个态叠加态的单个铷原子送入。这两种量子态都会引起微波场相位的移动,但移动量不同,因此场本身也处于两种量子态的叠加状态。由于光子在腔镜缺陷上的散射,腔场失去了与环境的相干性。


[math]\displaystyle{ \rho_\text{sys} = \operatorname{Tr}_\text{env}\Big(\sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \otimes |\epsilon_i\rang \lang\epsilon_j|\Big) = \sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \lang\epsilon_j|\epsilon_i\rang = \sum_{i,j} \psi_i \psi_j^* |i\rang \lang j| \delta_{ij} = \sum_i |\psi_i|^2 |i\rang \lang i|. }[/math]

Haroche and his colleagues measured the resulting decoherence via correlations between the states of pairs of atoms sent through the cavity with various time delays between the atoms.

Haroche 和他的同事们通过在原子之间有不同时间延迟的成对原子的态之间的相关性来测量由此产生的退相干。


Similarly, the final reduced density matrix after the transition will be

同样,过渡后的最终约化密度矩阵

In July 2011, researchers from University of British Columbia and University of California, Santa Barbara were able to reduce environmental decoherence rate "to levels far below the threshold necessary for quantum information processing" by applying high magnetic fields in their experiment.

2011年7月,来自不列颠哥伦比亚大学和加州大学圣巴巴拉分校的研究人员在他们的实验中应用强磁场,将环境退相干速率降低到“远低于量子信息处理所需的阈值”。

[math]\displaystyle{ \sum_j |\phi_j|^2 |j\rang \lang j|. }[/math]


In August 2020 scientists reported that that ionizing radiation from environmental radioactive materials and cosmic rays may substantially limit the coherence times of qubits if they aren't shielded adequately which may be critical for realizing fault-tolerant superconducting quantum computers in the future.

2020年8月,科学家们报告说,如果量子比特没有得到足够的屏蔽,那么来自环境中放射性物质和宇宙射线的电离辐射可能会大大限制量子比特的相干时间,而这对于未来实现容错的超导量子计算机至关重要。

The transition probability will then be given as

跃迁概率如下所示

[math]\displaystyle{ \operatorname{prob}_\text{after}(\psi \to \phi) = \sum_{i,j} |\psi_i|^2 |\phi_j|^2 \lang j|i\rang \lang i|j\rang = \sum_i |\psi_i^* \phi_i|^2, }[/math]

Criticism of the adequacy of decoherence theory to solve the measurement problem has been expressed by Anthony Leggett: "I hear people murmur the dreaded word "decoherence". But I claim that this is

对于解决测量问题的脱相干理论的充分性的批评,安东尼 · 莱格特曾经表达过: “我听到人们低声抱怨可怕的词语“脱相干”。但我认为这是


a major red herring". Concerning the experimental relevance of decoherence theory, Leggett has stated: "Let us now try to assess the decoherence argument. Actually, the most economical tactic at this point would be to go directly to the results of the next section, namely that it is experimentally refuted! However, it is interesting to spend a moment enquiring why it was reasonable to anticipate this in advance of the actual experiments. In fact, the argument contains several major loopholes".

”一条主要的红鲱鱼”。关于退相干理论的实验相关性,莱格特说: “现在让我们试着评估退相干的论点。实际上,在这一点上最经济的策略是直接进入下一部分的结果,也就是说它被实验性地驳斥了!然而,有趣的是花一点时间去探究为什么在实际实验之前就预测到这一点是合理的。事实上,这一论点存在几个重大漏洞”。

which has no contribution from the interference terms

没有干扰项的影响

[math]\displaystyle{ \sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i. }[/math]

Before an understanding of decoherence was developed, the Copenhagen interpretation of quantum mechanics treated wave-function collapse as a fundamental, a priori process. Decoherence provides an explanatory mechanism for the appearance of wave function collapse and was first developed by David Bohm in 1952, who applied it to Louis DeBroglie's pilot-wave theory, producing Bohmian mechanics, the first successful hidden-variables interpretation of quantum mechanics. Decoherence was then used by Hugh Everett in 1957 to form the core of his many-worlds interpretation. However, decoherence was largely ignored for many years (with the exception of Zeh's work), did decoherent-based explanations of the appearance of wave-function collapse become popular, with the greater acceptance of the use of reduced density matrices. — beyond the realm of measurement. Of course, by definition, the claim that a merged but unmeasurable wave function still exists cannot be proven experimentally. Decoherence explains why a quantum system begins to obey classical probability rules after interacting with its environment (due to the suppression of the interference terms when applying Bohm's probability rules to the system).

在理解退相干之前,哥本哈根诠释的量子力学将波函数崩塌作为一个基本的先验过程。退相干为波函数崩塌的出现提供了一种解释机制,首先由 David Bohm 在1952年提出,他把退相干应用到 Louis DeBroglie 的导波理论中,产生了玻姆力学,第一个成功地解释了量子力学的隐变量。1957年,Hugh Everett 使用退相干技术,形成了他的《多世界诠释的核心。然而,多年来退相干基本上被忽略了(Zeh 的工作除外) ,随着降低密度矩阵的使用得到更广泛的接受,退相干基于波函数崩塌现象的解释变得流行起来。超越了测量的范围。当然,根据定义,声称一个合并但不可测量的波函数仍然存在无法通过实验证明。退相干解释了为什么一个量子系统在与其环境相互作用后开始遵守经典概率规则(这是因为在系统中应用玻姆的概率规则时,干涉项被抑制)。


The density-matrix approach has been combined with the Bohmian approach to yield a reduced-trajectory approach, taking into account the system reduced density matrix and the influence of the environment.[11]

密度矩阵法与 Bohmian法相结合,得出了“简化轨迹法”,同时考虑了系统简化密度矩阵和环境的影响。[12]

Operator-sum representation运算符和表示法

Consider a system S and environment (bath) B, which are closed and can be treated quantum-mechanically. Let [math]\displaystyle{ \mathcal H_S }[/math] and [math]\displaystyle{ \mathcal H_B }[/math] be the system's and bath's Hilbert spaces respectively. Then the Hamiltonian for the combined system is

考虑一个系统“S”和环境(浴)“B”,它们是封闭的,可以用量子力学来处理。设[math]\displaystyle{ \mathcal H\u S }[/math][math]\displaystyle{ \mathcal H\u B }[/math]分别为系统和bath的Hilbert空间。然后给出了组合系统的哈密顿量

[math]\displaystyle{ \hat{H} = \hat H_S \otimes \hat I_B + \hat I_S \otimes \hat H_B + \hat H_I, }[/math]


where [math]\displaystyle{ \hat H_S, \hat H_B }[/math] are the system and bath Hamiltonians respectively, [math]\displaystyle{ \hat H_I }[/math] is the interaction Hamiltonian between the system and bath, and [math]\displaystyle{ \hat I_S, \hat I_B }[/math] are the identity operators on the system and bath Hilbert spaces respectively. The time-evolution of the density operator of this closed system is unitary and, as such, is given by

其中,[math]\displaystyle{ \hat H\u S、\lt hat H\u B }[/math]分别是系统和bath哈密顿量,[math]\displaystyle{ \hat H\u I }[/math]是系统和bath Hilbert空间之间的相互作用哈密顿量,[math]\displaystyle{ \hat I\u S、\lt hat I\u B }[/math]分别是系统和bath Hilbert空间上的恒等算子。这个封闭系统的密度算子的时间演化是幺正的,因此,由下式给出

[math]\displaystyle{ \rho_{SB}(t) = \hat U(t) \rho_{SB}(0) \hat U^\dagger(t), }[/math]


where the unitary operator is [math]\displaystyle{ \hat U = e^{-i\hat{H}t/\hbar} }[/math]. If the system and bath are not entangled initially, then we can write [math]\displaystyle{ \rho_{SB} = \rho_S \otimes \rho_B }[/math]. Therefore, the evolution of the system becomes

其中酉算子是[math]\displaystyle{ \hat U=e^{-i\hat{H}t/\hbar} }[/math]。如果系统和浴最初不是[[量子纠缠],那么我们可以写[math]\displaystyle{ \rho{SB}=\rho\u S\otimes\rho\u B }[/math]。因此,系统的演化就变得

[math]\displaystyle{ \rho_{SB}(t) = \hat U (t)[\rho_S(0) \otimes \rho_B(0)] \hat U^\dagger(t). }[/math]


The system–bath interaction Hamiltonian can be written in a general form as

系统-槽相互作用哈密顿量可以用一般形式写成

[math]\displaystyle{ \hat H_I = \sum_i \hat S_i \otimes \hat B_i, }[/math]


where [math]\displaystyle{ \hat S_i \otimes \hat B_i }[/math] is the operator acting on the combined system–bath Hilbert space, and [math]\displaystyle{ \hat S_i, \hat B_i }[/math] are the operators that act on the system and bath respectively. This coupling of the system and bath is the cause of decoherence in the system alone. To see this, a partial trace is performed over the bath to give a description of the system alone:

其中,[math]\displaystyle{ \hat S\u i \otimes\hat B\u i }[/math]是作用于组合系统-bath-Hilbert空间的算子,[math]\displaystyle{ \hat S\u i,\hat B\u i }[/math]是分别作用于系统和bath的算子。系统和镀液的这种耦合是系统本身退相干的原因。为了了解这一点,在镀液上进行部分跟踪,以单独描述系统:

[math]\displaystyle{ \rho_S(t) = \operatorname{Tr}_B\big[\hat U(t)[\rho_S(0) \otimes \rho_B(0)] \hat U^\dagger(t)\big]. }[/math]


[math]\displaystyle{ \rho_S(t) }[/math] is called the reduced density matrix and gives information about the system only. If the bath is written in terms of its set of orthogonal basis kets, that is, if it has been initially diagonalized, then [math]\displaystyle{ \textstyle\rho_B(0) = \sum_j a_j |j\rangle \langle j| }[/math]. Computing the partial trace with respect to this (computational) basis gives

[math]\displaystyle{ \rho\u S(t) }[/math]称为“约化密度矩阵”,仅给出有关系统的信息。如果用一组正交基ket来写这个bath,也就是说,如果它最初是对角化的,那么[math]\displaystyle{ \textstyle\rho\u B(0)=\sum\u j a\u j | j\rangle\langle j | }[/math]。计算关于这个(计算)基础的部分轨迹

| first = Maximilian | last = Schlosshauer

2012年10月11日


| year = 2007

2007年

[math]\displaystyle{ \rho_S(t) = \sum_l \hat A_l \rho_S(0) \hat A^\dagger_l, }[/math]
| title = Decoherence and the Quantum-to-Classical Transition

| title = 退相干和量子到经典的跃迁


| edition = 1st

1st

where [math]\displaystyle{ \hat A_l, \hat A^\dagger_l }[/math] are defined as the Kraus operators and are represented as

| location = Berlin/Heidelberg

| location = Berlin/Heidelberg


| publisher = Springer

| publisher = Springer

[math]\displaystyle{ \hat A_l = \sqrt{a_j} \langle k| \hat U |j\rangle. }[/math]
}}
}}


This is known as the operator-sum representation (OSR). A condition on the Kraus operators can be obtained by using the fact that [math]\displaystyle{ \operatorname{Tr}[\rho_S(t)] = 1 }[/math]; this then gives

| first = E. | last = Joos

第一个 = e | 最后一个 = Joos


| year = 2003

2003年

[math]\displaystyle{ \sum_l \hat A^\dagger_l \hat A_l = \hat I_S. }[/math]
| title = Decoherence and the Appearance of a Classical World in Quantum Theory

量子理论中的退相干和经典世界的出现


| edition = 2nd

2nd

This restriction determines whether decoherence will occur or not in the OSR. In particular, when there is more than one term present in the sum for [math]\displaystyle{ \rho_S(t) }[/math], then the dynamics of the system will be non-unitary, and hence decoherence will take place.

这个限制决定了OSR中是否会发生退相干。特别地,当在[math]\displaystyle{ \rhos(t) }[/math]的和中存在多个项时,系统的动力学将是非幺正的,因此将发生退相干。

| location = Berlin

地点: 柏林


| publisher = Springer

| publisher = Springer

Semigroup approach半群方法

|display-authors=etal}}

| display-authors = etal }

A more general consideration for the existence of decoherence in a quantum system is given by the master equation, which determines how the density matrix of the system alone evolves in time (see also the Belavkin equation

“主方程”给出了量子系统中存在退相干的更一般的考虑,它确定了“系统本身”的密度矩阵如何随时间演化(另见贝拉夫金方程[13][14][15] for the evolution under continuous measurement). This uses the Schrödinger picture, where evolution of the state (represented by its density matrix) is considered. The master equation is

| issue=2004

2004年


| arxiv=quant-ph/0312059

| arxiv=quant-ph/0312059

[math]\displaystyle{ \rho'_S(t) = \frac{-i}{\hbar} \big[\tilde H_S, \rho_S(t)\big] + L_D \big[\rho_S(t)\big], }[/math]

| bibcode=2004RvMP...76.1267S| s2cid = 7295619

2004RvMP... 76.1267 s | s2cid = 7295619


}}
}}

where [math]\displaystyle{ \tilde H_S = H_S + \Delta }[/math] is the system Hamiltonian [math]\displaystyle{ H_S }[/math] along with a (possible) unitary contribution [math]\displaystyle{ \Delta }[/math] from the bath, and [math]\displaystyle{ L_D }[/math] is the Lindblad decohering term.[5] The Lindblad decohering term is represented as

[math]\displaystyle{ L_D\big[\rho_S(t)\big] = \frac{1}{2} \sum_{\alpha, \beta = 1}^M b_{\alpha\beta} \Big(\big[\mathbf F_\alpha, \rho_S(t)\mathbf F^\dagger_\beta\big] + \big[\mathbf F_\alpha \rho_S(t), \mathbf F^\dagger_\beta\big]\Big). }[/math]


The [math]\displaystyle{ \{\mathbf{F}_\alpha\}_{\alpha=1}^M }[/math] are basis operators for the M-dimensional space of bounded operators that act on the system Hilbert space [math]\displaystyle{ \mathcal H_S }[/math] and are the error generators.[16] The matrix elements [math]\displaystyle{ b_{\alpha\beta} }[/math] represent the elements of a positive semi-definite Hermitian matrix; they characterize the decohering processes and, as such, are called the noise parameters.[16] The semigroup approach is particularly nice, because it distinguishes between the unitary and decohering (non-unitary) processes, which is not the case with the OSR. In particular, the non-unitary dynamics are represented by [math]\displaystyle{ L_D }[/math], whereas the unitary dynamics of the state are represented by the usual Heisenberg commutator. Note that when [math]\displaystyle{ L_D\big[\rho_S(t)\big] = 0 }[/math], the dynamical evolution of the system is unitary. The conditions for the evolution of the system density matrix to be described by the master equation are:[5]

  1. the evolution of the system density matrix is determined by a one-parameter semigroup,
  1. the evolution is "completely positive" (i.e. probabilities are preserved),
  1. the system and bath density matrices are initially decoupled.
  1. 系统密度矩阵的演化由一个单参数半群决定,
  1. 进化是“完全积极的”(即概率被保留),
  1. 系统和镀液密度矩阵“最初”解耦。

Examples of non-unitary modelling of decoherence退相干的非幺正建模实例

Decoherence can be modelled as a non-unitary process by which a system couples with its environment (although the combined system plus environment evolves in a unitary fashion).[5] Thus the dynamics of the system alone, treated in isolation, are non-unitary and, as such, are represented by irreversible transformations acting on the system's Hilbert space [math]\displaystyle{ \mathcal{H} }[/math]. Since the system's dynamics are represented by irreversible representations, then any information present in the quantum system can be lost to the environment or heat bath. Alternatively, the decay of quantum information caused by the coupling of the system to the environment is referred to as decoherence.[4] Thus decoherence is the process by which information of a quantum system is altered by the system's interaction with its environment (which form a closed system), hence creating an entanglement between the system and heat bath (environment). As such, since the system is entangled with its environment in some unknown way, a description of the system by itself cannot be made without also referring to the environment (i.e. without also describing the state of the environment).

Decoherence可以被建模为一个非 unity过程,通过这个过程,一个系统与其环境耦合(尽管组合的系统与环境以一种统一的方式演化)。引用错误:无效<ref>标签;name属性非法,可能是内容过长因此,孤立地处理系统的 Dynamics是非统一的,因此,由作用于系统Hilbert空间[math]\displaystyle{ \mathcal{H} }[/math]上的不可逆变换表示。由于系统的动力学是用不可逆的表示来表示的,那么量子系统中的任何信息都可能丢失到环境中或热浴。或者,由系统与环境耦合引起的量子信息衰减称为退相干。因此,退相干是指系统与其环境(形成一个封闭系统)的相互作用改变量子系统信息的过程,因此在系统和热浴(环境)之间产生了纠缠。因此,由于系统以某种未知的方式与其环境纠缠在一起,因此在不同时参考环境(即,不同时描述环境的状态)的情况下,不能单独描述系统。

Rotational decoherence旋转退相干

Consider a system of N qubits that is coupled to a bath symmetrically. Suppose this system of N qubits undergoes a rotation around the [math]\displaystyle{ |{\uparrow}\rangle \langle{\uparrow}|, |{\downarrow}\rangle \langle{\downarrow}| }[/math] [math]\displaystyle{ \big(|0\rangle \langle0|, |1\rangle \langle1|\big) }[/math] eigenstates of [math]\displaystyle{ \hat{J_z} }[/math]. Then under such a rotation, a random phase [math]\displaystyle{ \phi }[/math] will be created between the eigenstates [math]\displaystyle{ |0\rangle }[/math], [math]\displaystyle{ |1\rangle }[/math] of [math]\displaystyle{ \hat{J_z} }[/math]. Thus these basis qubits [math]\displaystyle{ |0\rangle }[/math] and [math]\displaystyle{ |1\rangle }[/math] will transform in the following way:

考虑一个对称地耦合到槽中的“N”量子位系统。假设这个“N”量子位系统绕着[math]\displaystyle{ {\uparrow}\rangle\langle{\uparrow}}、{\downarow}\rangle\langle{\downarow}{/math\gt \lt math\gt \big(| 0\rangle\langle0 |、| 1\rangle\langle1 |\big) }[/math]本征态旋转。然后在这样的旋转下,将在[math]\displaystyle{ \hat{J\u z} }[/math]的本征态[math]\displaystyle{ \0\rangle }[/math][math]\displaystyle{ \rangle }[/math]之间创建随机相位[math]\displaystyle{ \phi }[/math]。因此,这些基量子位[math]\displaystyle{ |0\rangle }[/math][math]\displaystyle{ |1\rangle }[/math]将按以下方式变换:

[math]\displaystyle{ |0\rangle \to |0\rangle, \quad |1\rangle \to e^{i\phi} |1\rangle. }[/math]


This transformation is performed by the rotation operator

此变换由旋转操作符执行

Decoherence

退相干


Category:Articles containing video clips

类别: 包含视频剪辑的文章

<math>R_z(\phi) =

Category:1970 introductions

类别: 1970年引言


This page was moved from wikipedia:en:Quantum decoherence. Its edit history can be viewed at 量子退相干性/edithistory

  1. H. Dieter Zeh, "On the Interpretation of Measurement in Quantum Theory", Foundations of Physics, vol. 1, pp. 69–76, (1970).
  2. Schlosshauer, Maximilian (2005). "Decoherence, the measurement problem, and interpretations of quantum mechanics". Reviews of Modern Physics. 76 (4): 1267–1305. arXiv:quant-ph/0312059. Bibcode:2004RvMP...76.1267S. doi:10.1103/RevModPhys.76.1267. Unknown parameter |s2cid= ignored (help)
  3. Joos and Zeh (1985) state ‘'Of course no unitary treatment of the time dependence can explain why only one of these dynamically independent components is experienced.'’ And in a recent Decoherence was first introduced in 1970 by the German physicist H. Dieter Zeh and has been a subject of active research since the 1980s. Decoherence has been developed into a complete framework, but it does not solve the measurement problem, as the founders of decoherence theory admit in their seminal papers. 退相干最早是在1970年由德国物理学家 h. Dieter Zeh 提出的,自20世纪80年代以来一直是一个活跃的研究课题。退相干已经发展成为一个完整的框架,但它并没有解决测量问题,正如退相干理论的创始人在他们的开创性论文中所承认的那样。 review on decoherence, Joos (1999) states ‘'Does decoherence solve the measurement problem? Clearly not. What decoherence tells us is that certain objects appear classical when observed. But what is an observation? At some stage we still have to apply the usual probability rules of quantum theory.'’Adler, Stephen L. (2003). "Why decoherence has not solved the measurement problem: a response to P.W. Anderson". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 34 (1): 135–142. arXiv:quant-ph/0112095. Bibcode:2003SHPMP..34..135A. doi:10.1016/S1355-2198(02)00086-2. Unknown parameter |s2cid= ignored (help)
  4. 4.0 4.1 4.2 Bacon, D. (2001). "Decoherence, control, and symmetry in quantum computers". arXiv:quant-ph/0305025.
  5. 5.0 5.1 5.2 5.3 5.4 Lidar, Daniel A.; Whaley, K. Birgitta (2003). "Decoherence-Free Subspaces and Subsystems". In Benatti, F.; Floreanini, R.. Irreversible Quantum Dynamics. Springer Lecture Notes in Physics. 622. Berlin. pp. 83–120. arXiv:quant-ph/0301032. Bibcode 2003LNP...622...83L. doi:10.1007/3-540-44874-8_5. ISBN 978-3-540-40223-7. 
  6. 6.00 6.01 6.02 6.03 6.04 6.05 6.06 6.07 6.08 6.09 6.10 6.11 Zurek, Wojciech H. (2003). "Decoherence, einselection, and the quantum origins of the classical". Reviews of Modern Physics. 75 (3): 715. arXiv:quant-ph/0105127. Bibcode:2003RvMP...75..715Z. doi:10.1103/revmodphys.75.715. Unknown parameter |s2cid= ignored (help)
  7. 引用错误:无效<ref>标签;未给name属性为“zurek03”的引用提供文字
  8. Wojciech H. Zurek, "Decoherence and the transition from quantum to classical", Physics Today, 44, pp. 36–44 (1991).
  9. Zurek [math]\displaystyle{ \mathcal H = \mathcal H_A \otimes \mathcal H_\epsilon. }[/math] 数学,数学,数学,数学, Wojciech (2002). [https://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf where [math]\displaystyle{ |\psi_1\rangle }[/math] and [math]\displaystyle{ |\psi_2\rangle }[/math] are orthogonal, and there is no entanglement initially. Also, choose an orthonormal basis [math]\displaystyle{ \{ |e_i\rangle \}_i }[/math] for [math]\displaystyle{ \mathcal H_A }[/math]. (This could be a "continuously indexed basis" or a mixture of continuous and discrete indexes, in which case we would have to use a rigged Hilbert space and be more careful about what we mean by orthonormal, but that's an inessential detail for expository purposes.) Then, we can expand 其中 < math > "Decoherence and the Transition from Quantum to Classical—Revisited This is a reasonably good approximation in the case where A and [math]\displaystyle{ \epsilon }[/math] are relatively independent (e.g. there is nothing like parts of A mixing with parts of [math]\displaystyle{ \epsilon }[/math] or conversely). The point is, the interaction with the environment is for all practical purposes unavoidable (e.g. even a single excited atom in a vacuum would emit a photon, which would then go off). Let's say this interaction is described by a unitary transformation U acting upon [math]\displaystyle{ \mathcal H }[/math]. Assume that the initial state of the environment is [math]\displaystyle{ |\text{in}\rangle }[/math], and the initial state of A is the superposition state 在 a 和 < math > epsilon </math > 相对独立的情况下,这是一个相当好的近似值。没有什么比 a 的某些部分和 e 的某些部分混合更好的了。关键是,与环境的相互作用在实际中是不可避免的(例如:。即使是真空中的单个受激原子也会发射出光子,然后光子就会爆炸)。我们假设这种相互作用是由一个幺正来描述的。假设环境的初始状态为 < math >"] Check |url= value (help). Los Alamos Science. 27 [math]\displaystyle{ c_1 |\psi_1\rangle + c_2 |\psi_2\rangle, }[/math] 1. arXiv:[//arxiv.org/abs/quant-ph/0306072 [math]\displaystyle{ U\big(|\psi_1\rangle \otimes |\text{in}\rangle\big) }[/math] 大的( quant-ph/0306072 '"`UNIQ--math-00000101-QINU`"' 大的(] Check |arxiv= value (help). Bibcode:2003quant.ph..6072Z. Text " text { in } rangle big) " ignored (help); Text " text { in } rangle </math > ,并且 a 的初始状态为叠加状态 " ignored (help); Text " psi 1 rangle + c 2 " ignored (help); Text " psi _ 1和 < math > " ignored (help); Text " psi _ 2和 </math > 是正交的,最初没有纠缠。另外,选择一个标准正交基。(这可以是一个“连续索引基础” ,或者是连续索引和离散索引的混合,在这种情况下,我们必须使用结构希尔伯特空间 ,并且更加小心我们所说的标准正交法,但是这对于说明性的目的来说是一个不重要的细节。)然后,我们可以扩张 " ignored (help); Text " psi 2 rangle,</math > " ignored (help); Text " psi _ 1 rangle o times " ignored (help); line feed character in |last= at position 6 (help); line feed character in |title= at position 67 (help); line feed character in |volume= at position 3 (help); line feed character in |url= at position 55 (help); line feed character in |arxiv= at position 17 (help)
  10. 10.0 10.1 E. Joos and H. D. Zeh, "The emergence of classical properties through interaction with the environment", Zeitschrift für Physik B, 59(2), pp. 223–243 (June 1985): eq. 1.2.
  11. A. S. Sanz, F. Borondo: A quantum trajectory description of decoherence, quant-ph/0310096v5.
  12. A. S. Sanz, F. Borondo: A quantum trajectory description of decoherence, quant-ph/0310096v5.
  13. Omnes 第一 = r, R. (1999 1999年). "Understanding Quantum Mechanics 文章标题: 了解量子力学". Physics Letters A. Princeton: 普林斯顿大学出版社. 140 (7–8): 355–358. Unknown parameter |位置= ignored (help); Unknown parameter |最后= ignored (help); line feed character in |last= at position 6 (help); line feed character in |year= at position 5 (help); line feed character in |title= at position 32 (help); More than one of |author= and |last= specified (help); Check date values in: |year= (help) }} | year = 1989 | doi = 10.1016/0375-9601(89)90066-2 | arxiv = quant-ph/0512136|bibcode = 1989PhLA..140..355B | s2cid = 6083856 | first = Maximilian | last = Schlosshauer 2012年10月11日 }}
  14. Howard J. Carmichael (2005年2月23日). An Open Systems Approach to Quantum Optics. 76. Berlin Heidelberg New-York: Springer-Verlag. pp. 1267-1305. doi:10.1103/RevModPhys. 76.1267. 
  15. 模板:Cite techreport
  16. 16.0 16.1 * Lidar, D. A.; Chuang, I. L.; Whaley, K. B. (1998). "Decoherence-Free Subspaces for Quantum Computation". Physical Review Letters. 81 (12): 2594–2597. arXiv:quant-ph/9807004. Bibcode:1998PhRvL..81.2594L. doi:10.1103/PhysRevLett.81.2594. Unknown parameter |s2cid= ignored (help)