# 量子退相干性

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.

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.

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.

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.

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]

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.

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.

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".

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".

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.

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.

## 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.

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.

Rabi oscillations]]

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 $\displaystyle{ \psi(x_1, x_2, \dots, x_N) }$, 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 $\displaystyle{ \psi }$ lives in the mathematical structure of a Hilbert space. Aside from these differences, however, the rough analogy holds.

An N-particle system can be represented in non-relativistic quantum mechanics by a wave function $\displaystyle{ \psi(x_1, x_2, \dots, x_N) }$, 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 $\displaystyle{ \psi }$ lives in the mathematical structure of a Hilbert space. Aside from these differences, however, the rough analogy holds.

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.

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.

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.

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.

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

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

$\displaystyle{ |\psi\rang = \sum_i |i\rang \lang i |\psi\rang, }$


[数学][数学]

$\displaystyle{ |\psi\rang = \sum_i |i\rang \lang i |\psi\rang, }$

where the $\displaystyle{ |i\rang }$s form an einselected basis (environmentally induced selected eigenbasis

where the $\displaystyle{ |i\rang }$s form an einselected basis (environmentally induced selected eigenbasis[6]), and let the environment initially be in the state $\displaystyle{ |\epsilon\rang }$. 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.

$\displaystyle{ |\text{before}\rang = \sum_i |i\rang |\epsilon\rang \lang i|\psi\rang, }$

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


where $\displaystyle{ |i\rang |\epsilon\rang }$ is shorthand for the tensor product $\displaystyle{ |i\rang \otimes |\epsilon\rang }$. 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

#### 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

$\displaystyle{ \sum_j |\phi_j|^2 |j\rang \lang j|. }$


[数学，数学]

$\displaystyle{ |i\rang |\epsilon\rang }$ evolves into $\displaystyle{ |\epsilon_i\rang, }$

The transition probability will then be given as

and so

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


$\displaystyle{ |\text{before}\rang }$ evolves into $\displaystyle{ |\text{after}\rang = \sum_i |\epsilon_i\rang \lang i|\psi\rang. }$

which has no contribution from the interference terms

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

$\displaystyle{ \lang i|j\rang = \delta_{ij} }$:

$\displaystyle{ \sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i. }$


$\displaystyle{ \lang\epsilon_i|\epsilon_j\rang = \delta_{ij}. }$

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.

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

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

#### 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.

$\displaystyle{ \hat{H} = \hat H_S \otimes \hat I_B + \hat I_S \otimes \hat H_B + \hat H_I, }$


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

where $\displaystyle{ \hat H_S, \hat H_B }$ are the system and bath Hamiltonians respectively, $\displaystyle{ \hat H_I }$ is the interaction Hamiltonian between the system and bath, and $\displaystyle{ \hat I_S, \hat I_B }$ 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

$\displaystyle{ |i\rang |\epsilon\rang }$ evolves into the product $\displaystyle{ |i, \epsilon_i\rang = |i\rang |\epsilon_i\rang, }$

$\displaystyle{ \rho_{SB}(t) = \hat U(t) \rho_{SB}(0) \hat U^\dagger(t), }$


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

and so 于是

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

$\displaystyle{ |\text{before}\rang }$ evolves into $\displaystyle{ |\text{after}\rang = \sum_i |i, \epsilon_i\rang \lang i|\psi\rang. }$

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


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

In this case, unitarity demands that

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

$\displaystyle{ \lang i, \epsilon_i|j, \epsilon_j\rang = \lang i|j\rang \lang\epsilon_i|\epsilon_j\rang = \delta_{ij} \lang\epsilon_i|\epsilon_j\rang = \delta_{ij} \lang\epsilon_i|\epsilon_i\rang = \delta_{ij}, }$

$\displaystyle{ \hat H_I = \sum_i \hat S_i \otimes \hat B_i, }$


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

where $\displaystyle{ \hat S_i \otimes \hat B_i }$ is the operator acting on the combined system–bath Hilbert space, and $\displaystyle{ \hat S_i, \hat B_i }$ 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:

$\displaystyle{ \lang\epsilon_i|\epsilon_j\rang \approx \delta_{ij}. }$

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


“ 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.

$\displaystyle{ \rho_S(t) }$ 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 $\displaystyle{ \textstyle\rho_B(0) = \sum_j a_j |j\rangle \langle j| }$. 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 $\displaystyle{ |i\rang }$ were not an einselected basis, then the last condition is trivial, since the disturbed environment is not a function of $\displaystyle{ i }$, and we have the trivial disturbed environment basis $\displaystyle{ |\epsilon_j\rang = |\epsilon'\rang }$. 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.

$\displaystyle{ \rho_S(t) = \sum_l \hat A_l \rho_S(0) \hat A^\dagger_l, }$


[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 $\displaystyle{ \psi }$ to $\displaystyle{ \phi }$ before $\displaystyle{ \psi }$ 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:

where $\displaystyle{ \hat A_l, \hat A^\dagger_l }$ are defined as the Kraus operators and are represented as

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


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

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

The above expansion of the transition probability has terms that involve $\displaystyle{ i \ne j }$; 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.

$\displaystyle{ \sum_l \hat A^\dagger_l \hat A_l = \hat I_S. }$


[数学][数学][数学]

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

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 $\displaystyle{ \rho_S(t) }$, then the dynamics of the system will be non-unitary, and hence decoherence will take place.

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

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

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

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


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

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

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

or, equivalently, the decay of the purity

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 $\displaystyle{ \textstyle\sum_i }$ from inside of the modulus sign to outside. As a result, all the cross- or quantum interference-terms

.

.

$\displaystyle{ \sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i }$

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

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]

$\displaystyle{ U\big(|\psi_2\rangle \otimes |\text{in}\rangle\big) }$


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.

uniquely as

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

$\displaystyle{ \sum_i |e_i\rangle \otimes |f_{1i}\rangle }$


$\displaystyle{ \rho = |\text{before}\rang \lang\text{before}| = |\psi\rang \lang\psi| \otimes |\epsilon\rang \lang\epsilon|, }$

and

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

$\displaystyle{ \sum_i |e_i\rangle \otimes |f_{2i}\rangle }$


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:

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 $\displaystyle{ \mathcal H_\epsilon }$ such that $\displaystyle{ |f_{1i}\rangle }$ and $\displaystyle{ |f_{1j}\rangle }$ are all approximately orthogonal to a good degree if i ≠ j and the same thing for $\displaystyle{ |f_{2i}\rangle }$ and $\displaystyle{ |f_{2j}\rangle }$ and also for $\displaystyle{ |f_{1i}\rangle }$ and $\displaystyle{ |f_{2j}\rangle }$ for any i and j (the decoherence property).

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

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

Now the transition probability will be given as

$\displaystyle{ \sum_i \big(\langle f_{1i}|f_{1i}\rangle + \langle f_{2i}|f_{2i}\rangle\big) |e_i\rangle \langle e_i|, }$

$\displaystyle{ \sum_i \big(\langle f_{1i}|f_{1i}\rangle + \langle f_{2i}|f_{2i}\rangle\big) |e_i\rangle \langle e_i|, }$

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

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.

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

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

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

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.

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]

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。

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

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

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”。

$\displaystyle{ \sum_j |\phi_j|^2 |j\rang \lang j|. }$

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

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

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

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

$\displaystyle{ \sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i. }$

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).

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]

### 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 $\displaystyle{ \mathcal H_S }$ and $\displaystyle{ \mathcal H_B }$ be the system's and bath's Hilbert spaces respectively. Then the Hamiltonian for the combined system is

$\displaystyle{ \hat{H} = \hat H_S \otimes \hat I_B + \hat I_S \otimes \hat H_B + \hat H_I, }$

where $\displaystyle{ \hat H_S, \hat H_B }$ are the system and bath Hamiltonians respectively, $\displaystyle{ \hat H_I }$ is the interaction Hamiltonian between the system and bath, and $\displaystyle{ \hat I_S, \hat I_B }$ 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

$\displaystyle{ \rho_{SB}(t) = \hat U(t) \rho_{SB}(0) \hat U^\dagger(t), }$

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

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

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

$\displaystyle{ \hat H_I = \sum_i \hat S_i \otimes \hat B_i, }$

where $\displaystyle{ \hat S_i \otimes \hat B_i }$ is the operator acting on the combined system–bath Hilbert space, and $\displaystyle{ \hat S_i, \hat B_i }$ 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:

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

$\displaystyle{ \rho_S(t) }$ 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 $\displaystyle{ \textstyle\rho_B(0) = \sum_j a_j |j\rangle \langle j| }$. Computing the partial trace with respect to this (computational) basis gives

$\displaystyle{ \rho\u S（t） }$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$\displaystyle{ \textstyle\rho\u B（0）=\sum\u j a\u j | j\rangle\langle j | }$。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

$\displaystyle{ \rho_S(t) = \sum_l \hat A_l \rho_S(0) \hat A^\dagger_l, }$
| 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 $\displaystyle{ \hat A_l, \hat A^\dagger_l }$ are defined as the Kraus operators and are represented as

| location = Berlin/Heidelberg


| location = Berlin/Heidelberg

| publisher = Springer


| publisher = Springer

$\displaystyle{ \hat A_l = \sqrt{a_j} \langle k| \hat U |j\rangle. }$
}}

}}


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

| first = E. | last = Joos


| year = 2003


2003年

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


| 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 $\displaystyle{ \rho_S(t) }$, then the dynamics of the system will be non-unitary, and hence decoherence will take place.

| location = Berlin


| 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

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

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

2004RvMP... 76.1267 s | s2cid = 7295619

}}

}}


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

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

The $\displaystyle{ \{\mathbf{F}_\alpha\}_{\alpha=1}^M }$ are basis operators for the M-dimensional space of bounded operators that act on the system Hilbert space $\displaystyle{ \mathcal H_S }$ and are the error generators.[16] The matrix elements $\displaystyle{ b_{\alpha\beta} }$ 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 $\displaystyle{ L_D }$, whereas the unitary dynamics of the state are represented by the usual Heisenberg commutator. Note that when $\displaystyle{ L_D\big[\rho_S(t)\big] = 0 }$, 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 $\displaystyle{ \mathcal{H} }$. 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$\displaystyle{ \mathcal{H} }$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 $\displaystyle{ |{\uparrow}\rangle \langle{\uparrow}|, |{\downarrow}\rangle \langle{\downarrow}| }$ $\displaystyle{ \big(|0\rangle \langle0|, |1\rangle \langle1|\big) }$ eigenstates of $\displaystyle{ \hat{J_z} }$. Then under such a rotation, a random phase $\displaystyle{ \phi }$ will be created between the eigenstates $\displaystyle{ |0\rangle }$, $\displaystyle{ |1\rangle }$ of $\displaystyle{ \hat{J_z} }$. Thus these basis qubits $\displaystyle{ |0\rangle }$ and $\displaystyle{ |1\rangle }$ will transform in the following way:

$\displaystyle{ |0\rangle \to |0\rangle, \quad |1\rangle \to e^{i\phi} |1\rangle. }$

This transformation is performed by the rotation operator

Decoherence

Category:Articles containing video clips

[itex]R_z(\phi) =

Category:1970 introductions

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. Bacon, D. (2001). "Decoherence, control, and symmetry in quantum computers". arXiv:quant-ph/0305025.
5. 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. 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 $\displaystyle{ \mathcal H = \mathcal H_A \otimes \mathcal H_\epsilon. }$ 数学，数学，数学，数学, Wojciech (2002). [https://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf where $\displaystyle{ |\psi_1\rangle }$ and $\displaystyle{ |\psi_2\rangle }$ are orthogonal, and there is no entanglement initially. Also, choose an orthonormal basis $\displaystyle{ \{ |e_i\rangle \}_i }$ for $\displaystyle{ \mathcal H_A }$. (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 $\displaystyle{ \epsilon }$ are relatively independent (e.g. there is nothing like parts of A mixing with parts of $\displaystyle{ \epsilon }$ 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 $\displaystyle{ \mathcal H }$. Assume that the initial state of the environment is $\displaystyle{ |\text{in}\rangle }$, 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 $\displaystyle{ c_1 |\psi_1\rangle + c_2 |\psi_2\rangle, }$ 1. arXiv:[//arxiv.org/abs/quant-ph/0306072 $\displaystyle{ U\big(|\psi_1\rangle \otimes |\text{in}\rangle\big) }$ 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. 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. * 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)