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- Quantum decoherence

**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.

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 there is controversy as to whether it solves 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),^{[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 possibility of the collapse of the wave function in quantum mechanics. Decoherence does not generate *actual* wave-function collapse. It only provides a framework 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. With respect to the measurement problem, 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.

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.

*\psi(x*_{1,}*x*_{2,}...*,**x*_{N)}

*\psi*

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.^{[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.

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.

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

*|\psi\rang*=*\sum*_{i}*|i\rang**\lang**i**|\psi\rang,*

where the

*|i\rang*

*|\epsilon\rang*

*|before\rang*=*\sum*_{i}*|i\rang**|\epsilon\rang**\lang**i|\psi\rang,*

where

*|i\rang**|\epsilon\rang*

*|i\rang* ⊗ *|\epsilon\rang*

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

*|i\rang**|\epsilon\rang*

*|\epsilon*_{i\rang,}

and so

*|before\rang*

*|after\rang*=*\sum*_{i}*|\epsilon*_{i\rang}*\lang**i|\psi\rang.*

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

*\lang**i|j\rang*=*\delta*_{ij}

*\lang\epsilon*_{i|\epsilon}_{j\rang}=*\delta*_{ij}*.*

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

In an idealised measurement, the system disturbs the environment, but is itself undisturbed by the environment.In this case, each element of the basis interacts with the environment such that

*|i\rang**|\epsilon\rang*

*|i,**\epsilon*_{i\rang}=*|i\rang**|\epsilon*_{i\rang,}

and so

*|before\rang*

*|after\rang*=*\sum*_{i}*|i,**\epsilon*_{i\rang}*\lang**i|\psi\rang.*

In this case, unitarity demands that

*\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}*,*

where

*\lang**\epsilon*_{i}*|**\epsilon*_{i}*\rang*=1

*\lang\epsilon*_{i|\epsilon}_{j\rang} ≈ *\delta*_{ij}*.*

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.

Note that if the system basis

*|i\rang*

*i*

*|\epsilon*_{j\rang}=*|\epsilon'\rang*

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

*\psi*

*\phi*

*\psi*

*\operatorname{prob}*_{before(\psi}*\to**\phi)*=*\left|\lang\psi|\phi\rang\right|*^{2}=*\left|\sum*_{i}

* | |

\psi | |

i |

2 | |

\phi | |

i\right| |

=*\sum*_{i}

* | |

|\psi | |

i |

2 | |

\phi | |

i| |

+*\sum*_{ij;}

* | |

\psi | |

i |

*\psi*_{j}

* | |

\phi | |

j |

*\phi*_{i}*,*

where

*\psi*_{i}=*\lang**i|\psi\rang*

* | |

\psi | |

i |

=*\lang\psi|i\rang*

*\phi*_{i}=*\lang**i|\phi\rang*

The above expansion of the transition probability has terms that involve

*i**\ne**j*

To calculate the probability of observing the system making a quantum leap from

*\psi*

*\phi*

*\psi*

*|\epsilon*_{i\rang}

*\operatorname{prob}*_{after(\psi}*\to**\phi)*=*\sum*_{j}*\left|\langafter\right|**\phi,**\epsilon*_{j}*\rang|*^{2}=*\sum*_{j}*\left|\sum*_{i}

* | |

\psi | |

i |

*\lang**i,**\epsilon*_{i|\phi,}

2 | |

\epsilon | |

j\rang\right| |

=*\sum*_{j\left|\sum}_{i}

* | |

\psi | |

i |

*\phi*_{i}*\lang\epsilon*_{i|\epsilon}_{j\rang}*\right|*^{2.}

The internal summation vanishes when we apply the decoherence/einselection condition

*\lang\epsilon*_{i|\epsilon}_{j\rang} ≈ *\delta*_{ij}

*\operatorname{prob}*_{after(\psi}*\to**\phi)* ≈ *\sum*_{j}

* | |

|\psi | |

j |

2 | |

\phi | |

j| |

=*
\sum*_{i}

* | |

|\psi | |

i |

2. | |

\phi | |

i| |

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

style\sum_{i}

*\sum*_{ij;}

* | |

\psi | |

i |

*\psi*_{j}

* | |

\phi | |

j |

*\phi*_{i}

have vanished from the transition-probability calculation. The decoherence has irreversibly converted quantum behaviour (additive probability amplitudes) to classical behaviour (additive probabilities).^{[6]} ^{[7]} ^{[8]}

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

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.

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

*\rho*=*|before\rang**\langbefore|*=*|\psi\rang**\lang\psi|* ⊗ *|\epsilon\rang**\lang\epsilon|,*

where

*|\epsilon\rang*

*\rho*_{sys}=*\operatorname{Tr}*_{rm{env}(\rho)=}*|\psi\rang**\lang\psi|**\lang\epsilon|\epsilon\rang*=*|\psi\rang**\lang\psi|.*

Now the transition probability will be given as

*\operatorname{prob}*_{before(\psi}*\to**\phi)*=*\lang\phi|**\rho*_{sys}*|\phi\rang*=*\lang\phi|\psi\rang**\lang\psi|\phi\rang*=|*\lang\psi|\phi\rang*|^{2}=*\sum*_{i}

* | |

|\psi | |

i |

2 | |

\phi | |

i| |

+*\sum*_{ij;}

* | |

\psi | |

i |

*\psi*_{j}

* | |

\phi | |

j\phi |

_{i,}

where

*\psi*_{i}=*\lang**i|\psi\rang*

* | |

\psi | |

i |

=*\lang**\psi|i\rang*

*\phi*_{i}=*\lang**i|\phi\rang*

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

*\rho*=*|after\rang**\langafter|*=*\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|* ⊗ *|\epsilon*_{i\rang}*\lang\epsilon*_{j|.}

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):^{[9]}

*\rho*_{sys}=*\operatorname{Tr}*_{env(\sum}_{i,j}*\psi*_{i}

* | |

\psi | |

j |

*|i\rang**\lang**j|* ⊗ *|\epsilon*_{i\rang}*\lang\epsilon*_{j|)}=*\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}

2 | |

|\psi | |

i| |

*|i\rang**\lang**i|.*

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

*\sum*_{j}

2 | |

|\phi | |

j| |

*|j\rang**\lang**j|.*

The transition probability will then be given as

*\operatorname{prob}*_{after(\psi}*\to**\phi)*=*\sum*_{i,j}

2 | |

|\psi | |

i| |

2 | |

|\phi | |

j| |

*\lang**j|i\rang**\lang**i|j\rang*=*\sum*_{i}

* | |

|\psi | |

i |

2, | |

\phi | |

i| |

which has no contribution from the interference terms

*\sum*_{ij;}

* | |

\psi | |

i |

*\psi*_{j}

* | |

\phi | |

j |

*\phi*_{i.}

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.^{[10]}

Consider a system *S* and environment (bath) *B*, which are closed and can be treated quantum-mechanically. Let

l*H*_{S}

l*H*_{B}

*\hat{H}*=*\hat**H*_{S} ⊗ *\hat**I*_{B}+*\hat**I*_{S} ⊗ *\hat**H*_{B}+*\hat**H*_{I,}

where

*\hat**H*_{S,}*\hat**H*_{B}

*\hat**H*_{I}

*\hat**I*_{S,}*\hat**I*_{B}

*\rho*_{SB}*(t)*=*\hat**U(t)**\rho*_{SB}*(*0*)**\hat**U*^{\dagger(t),}

where the unitary operator is

*\hat**U*=*e*^{-i\hat{H}*t/\hbar}*

*\rho*_{SB}=*\rho*_{S} ⊗ *\rho*_{B}

*\rho*_{SB}*(t)*=*\hat**U**(t)[\rho*_{S(0)} ⊗ *\rho*_{B(0)]}*\hat**U*^{\dagger(t).}

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

*\hat**H*_{I}=*\sum*_{i}*\hat**S*_{i} ⊗ *\hat**B*_{i,}

where

*\hat**S*_{i} ⊗ *\hat**B*_{i}

*\hat**S*_{i,}*\hat**B*_{i}

*\rho*_{S(t)}=*\operatorname{Tr}*_{B[\hat}*U(t)[\rho*_{S(0)} ⊗ *\rho*_{B(0)]}*\hat**U*^{\dagger(t)].}

*\rho*_{S(t)}

style\rho_{B(0)}=*\sum*_{j}*a*_{j}*|j\rangle**\langle**j|*

*\rho*_{S(t)}=*\sum*_{l}*\hat**A*_{l}*\rho*_{S(0)}*\hat*

\dagger | |

A | |

l, |

where

*\hat**A*_{l,}*\hat*

\dagger | |

A | |

l |

*l*

*k*

*j*

*\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

*\operatorname{Tr}[\rho*_{S(t)]}=1

*\sum*_{l}*\hat*

\dagger | |

A | |

l |

*\hat**A*_{l}=*\hat**I*_{S.}

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

*\rho*_{S(t)}

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^{[11]} ^{[12]} ^{[13]} 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

*\rho'*_{S(t)}=

-i | |

\hbar |

[*\tilde**H*_{S,}*\rho*_{S(t)]}+*L*_{D}[*\rho*_{S(t)],}

where

*\tilde**H*_{S}=*H*_{S}+*\Delta*

*H*_{S}

*\Delta*

*L*_{D}

*L*_{D[\rho}_{S(t)]}=

1 | |

2 |

M | |

\sum | |

\alpha,\beta=1 |

*b*_{\alpha\beta}([F_{\alpha,}

\dagger | |

\rho | |

\beta] |

+[F_{\alpha}*\rho*_{S(t),}

\dagger | |

F | |

\beta]). |

The

*\{F*_{\alpha\}}

M | |

\alpha=1 |

l*H*_{S}

*b*_{\alpha\beta}

*L*_{D}

*L*_{D[\rho}_{S(t)]}=0

- the evolution of the system density matrix is determined by a one-parameter semigroup,
- the evolution is "completely positive" (i.e. probabilities are preserved),
- the system and bath density matrices are
*initially*decoupled.

l{H}

Consider a system of *N* qubits that is coupled to a bath symmetrically. Suppose this system of *N* qubits undergoes a rotation around the

*|{\uparrow}\rangle**\langle{\uparrow}|,**|{\downarrow}\rangle**\langle{\downarrow}|*

(*|*0*\rangle**\langle*0*|,**|*1*\rangle**\langle*1*|*)

*\hat{J*_{z}}

*\phi*

*|*0*\rangle*

*|*1*\rangle*

*\hat{J*_{z}}

*|*0*\rangle*

*|*1*\rangle*

*|*0*\rangle**\to**|*0*\rangle,* *|*1*\rangle**\to**e*^{i\phi}*|*1*\rangle.*

This transformation is performed by the rotation operator

*R*_{z(\phi)}=*
\begin{pmatrix}*1*&*0*\* *
*0*&**e*^{i\phi}*\end{pmatrix}
.*

Since any qubit in this space can be expressed in terms of the basis qubits, then all such qubits will be transformed under this rotation.Consider a qubit in a pure state

*|\psi*_{j\rangle}=*a**|*0_{j\rangle}+*b**|*1_{j\rangle}

*e*^{i\phi}

*\phi*

*\rho*_{j}=

1 | |

2\pi |

2\pi | |

\int\limits | |

0 |

*R*_{z(\phi)}*|\psi*_{j\rangle}*\langle\psi*_{j|R}

\dagger(\phi) | |

z |

*p(\phi)**d\phi,*

where

*p(\phi)*

*p(\phi)*

*p(\phi)*=*(*4*\pi\alpha)*^{-1/2}

| ||||

e |

*,*

then the density matrix is

*\rho*_{j}=*
\begin{pmatrix}
**|a|*^{2}*&**ab*^{*}*e*^{-\alpha}*\\
**a*^{*be}^{-\alpha}*&**|b|*^{2}*\end{pmatrix}
.*

Since the off-diagonal elements—the coherence terms—decay for increasing

*\alpha*

*|*0*\rangle**\langle*0*|,**|*1*\rangle**\langle*1*|*

**Depolarizing** is a non-unitary transformation on a quantum system which maps pure states to mixed states. This is a non-unitary process, because any transformation that reverses this process will map states out of their respective Hilbert space thus not preserving positivity (i.e. the original probabilities are mapped to negative probabilities, which is not allowed). The 2-dimensional case of such a transformation would consist of mapping pure states on the surface of the Bloch sphere to mixed states within the Bloch sphere. This would contract the Bloch sphere by some finite amount and the reverse process would expand the Bloch sphere, which cannot happen.

See main article: Quantum dissipation. **Dissipation** is a decohering process by which the populations of quantum states are changed due to entanglement with a bath. An example of this would be a quantum system that can exchange its energy with a bath through the interaction Hamiltonian. If the system is not in its ground state and the bath is at a temperature lower than that of the system's, then the system will give off energy to the bath, and thus higher-energy eigenstates of the system Hamiltonian will decohere to the ground state after cooling and, as such, will all be non-degenerate. Since the states are no longer degenerate, they are not distinguishable, and thus this process is irreversible (non-unitary).

Decoherence represents an extremely fast process for macroscopic objects, since these are interacting with many microscopic objects, with an enormous number of degrees of freedom, in their natural environment. The process is needed if we are to understand why we tend not to observe quantum behaviour in everyday macroscopic objects and why we do see classical fields emerge from the properties of the interaction between matter and radiation for large amounts of matter. The time taken for off-diagonal components of the density matrix to effectively vanish is called the **decoherence time**. It is typically extremely short for everyday, macroscale processes.^{[6]} ^{[7]} ^{[8]} A modern basis-independent definition of the decoherence time relies on the short-time behavior of the fidelity between the initial and the time-dependent state^{[15]} or, equivalently, the decay of the purity.^{[16]}

We assume for the moment that the system in question consists of a subsystem *A* being studied and the "environment"

*\epsilon*

l*H*_{A}

l*H*_{\epsilon}

*\epsilon*

l*H*=l*H*_{A} ⊗ l*H*_{\epsilon.}

This is a reasonably good approximation in the case where *A* and

*\epsilon*

*\epsilon*

l*H*

*|in\rangle*

*c*_{1}*|\psi*_{1\rangle}+*c*_{2}*|\psi*_{2\rangle,}

where

*|\psi*_{1\rangle}

*|\psi*_{2\rangle}

*\{**|e*_{i\rangle}*\}*_{i}

l*H*_{A}

*U*(*|\psi*_{1\rangle} ⊗ *|in\rangle*)

and

*U*(*|\psi*_{2\rangle} ⊗ *|in\rangle*)

uniquely as

*\sum*_{i}*|e*_{i\rangle} ⊗ *|f*_{1i}*\rangle*

and

*\sum*_{i}*|e*_{i\rangle} ⊗ *|f*_{2i}*\rangle*

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

l*H*_{\epsilon}

*|f*_{1i}*\rangle*

*|f*_{1j}*\rangle*

*|f*_{2i}*\rangle*

*|f*_{2j}*\rangle*

*|f*_{1i}*\rangle*

*|f*_{2j}*\rangle*

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

*\sum*_{i}(*\langle**f*_{1i}*|f*_{1i}*\rangle*+*\langle**f*_{2i}*|f*_{2i}*\rangle*)*|e*_{i\rangle}*\langle**e*_{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**.

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.^{[17]}

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.^{[18]} 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.

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.

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.^{[19]} ^{[20]} ^{[21]}

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.^{[22]} ^{[23]} ^{[24]}

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 isa major red herring".^{[25]} 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".^{[26]}

Before an understanding of decoherence was developed, the Copenhagen interpretation of quantum mechanics treated wave-function collapse as a fundamental, *a priori* process. Decoherence as a possible *explanatory mechanism* for the *appearance* of wave function collapse was first developed by David Bohm in 1952, who applied it to Louis DeBroglie's pilot-wave theory, producing Bohmian mechanics,^{[27]} ^{[28]} 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.^{[29]} However, decoherence was largely ignored for many years (with the exception of Zeh's work),^{[1]} and not until the 1980s^{[30]} ^{[31]} did decoherent-based explanations of the appearance of wave-function collapse become popular, with the greater acceptance of the use of reduced density matrices.^{[9]} ^{[7]} The range of decoherent interpretations have subsequently been extended around the idea, such as consistent histories. Some versions of the Copenhagen interpretation have been modified to include decoherence.

Decoherence does not claim to provide a mechanism for some actual wave-function collapse; rather it puts forth a reasonable framework for the appearance of wave-function collapse. The quantum nature of the system is simply "leaked" into the environment so that a total superposition of the wave function still exists, but exists – at least for all practical purposes^{[32]} — beyond the realm of measurement.^{[33]} Of course, by definition, the claim that a merged but unmeasurable wave function still exists cannot be proven experimentally. Decoherence is needed to understand 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 Born's probability rules to the system).

- Dephasing
- Dephasing rate SP formula
- Einselection
- Ghirardi–Rimini–Weber theory
- H. Dieter Zeh
- Interpretations of quantum mechanics
- Objective-collapse theory
- Partial trace
- Photon polarization
- Quantum coherence
- Quantum Darwinism
- Quantum entanglement
- Quantum superposition
- Quantum Zeno effect

- Book: Schlosshauer , Maximilian . 2007 . Decoherence and the Quantum-to-Classical Transition . 1st . Berlin/Heidelberg . Springer .
- Book: Joos , E. . 2003 . Decoherence and the Appearance of a Classical World in Quantum Theory . 2nd . Berlin . Springer . etal.
- Book: Omnes , R. . 1999 . Understanding Quantum Mechanics . Princeton . Princeton University Press .
- Zurek, Wojciech H. (2003). "Decoherence and the transition from quantum to classical – REVISITED", (An updated version of PHYSICS TODAY, 44:36–44 (1991) article)
- Maximilian . Schlosshauer . 23 February 2005 . Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics . Reviews of Modern Physics . 76 . 1267–1305 . 10.1103/RevModPhys.76.1267. 2004. quant-ph/0312059. 2004RvMP...76.1267S. 7295619 .
- J. J. Halliwell, J. Perez-Mercader, Wojciech H. Zurek, eds,
*The Physical Origins of Time Asymmetry*, Part 3: Decoherence, - Berthold-Georg Englert, Marlan O. Scully & Herbert Walther,
*Quantum Optical Tests of Complementarity*, Nature, Vol 351, pp 111–116 (9 May 1991) and (same authors)*The Duality in Matter and Light*Scientific American, pg 56–61, (December 1994). Demonstrates that complementarity is enforced, and quantum interference effects destroyed, by irreversible object-apparatus correlations, and not, as was previously popularly believed, by Heisenberg's uncertainty principle itself. - Mario Castagnino, Sebastian Fortin, Roberto Laura and Olimpia Lombardi,
*A general theoretical framework for decoherence in open and closed systems*, Classical and Quantum Gravity, 25, pp. 154002–154013, (2008). A general theoretical framework for decoherence is proposed, which encompasses formalisms originally devised to deal just with open or closed systems.

- Decoherence.info by Erich Joos
- http://plato.stanford.edu/entries/qm-decoherence/
- quant-ph/0312059. 10.1103/RevModPhys.76.1267. Decoherence, the measurement problem, and interpretations of quantum mechanics. 2005. Schlosshauer. Maximilian. Reviews of Modern Physics. 76. 4. 1267–1305. 7295619.
- quant-ph/0505070. Dass. Tulsi. Measurements and Decoherence. 2005.
- A detailed introduction from a graduate student's website at Drexel University
- Quantum Bug : Qubits might spontaneously decay in seconds
*Scientific American*(October 2005) - Quantum Decoherence and the Measurement Problem

- [H. Dieter Zeh]
- Schlosshauer . Maximilian . 2005 . Decoherence, the measurement problem, and interpretations of quantum mechanics . . 76 . 4 . 1267–1305 . 10.1103/RevModPhys.76.1267 . quant-ph/0312059 . 2004RvMP...76.1267S . 7295619 .
- 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 recentreview 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. . 34. 1. 135–142. quant-ph/0112095. 10.1016/S1355-2198(02)00086-2. 2003SHPMP..34..135A. 21040195.
- Bacon . D. . 2001 . Decoherence, control, and symmetry in quantum computers . quant-ph/0305025 .
- Book: Lidar . Daniel A. . K. Birgitta . Whaley . Irreversible Quantum Dynamics . Decoherence-Free Subspaces and Subsystems . Irreversible Quantum Dynamics . F. . Benatti . R. . Floreanini . 83–120 . Springer Lecture Notes in Physics . 622 . Berlin . 2003 . quant-ph/0301032 . 978-3-540-40223-7. 2003LNP...622...83L . 10.1007/3-540-44874-8_5 . 117748831 .
- Zurek . Wojciech H. . 2003 . Decoherence, einselection, and the quantum origins of the classical . quant-ph/0105127 . Reviews of Modern Physics . 75 . 3. 715 . 10.1103/revmodphys.75.715 . 2003RvMP...75..715Z. einselection . 14759237 .
- [Wojciech H. Zurek]
- Zurek . Wojciech . Decoherence and the Transition from Quantum to Classical—Revisited . Los Alamos Science . 27 . 2002 . 2003quant.ph..6072Z . quant-ph/0306072 .
- 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. - A. S. Sanz, F. Borondo:
*A quantum trajectory description of decoherence*, quant-ph/0310096v5. - V. P. Belavkin . A new wave equation for a continuous non-demolition measurement . Physics Letters A . 140 . 7–8 . 355–358 . 1989 . 10.1016/0375-9601(89)90066-2 . quant-ph/0512136. 1989PhLA..140..355B . 6083856 .
- Book: Howard J. Carmichael . An Open Systems Approach to Quantum Optics . Springer-Verlag . 1993 . Berlin Heidelberg New-York.
- Michel Bauer . Denis Bernard . Tristan Benoist . Iterated Stochastic Measurements . 1210.0425. 2012JPhA...45W4020B. 10.1088/1751-8113/45/49/494020.
- quant-ph/9807004. 10.1103/PhysRevLett.81.2594. 1998PhRvL..81.2594L. Decoherence-Free Subspaces for Quantum Computation. 1998. Lidar. D. A.. Chuang. I. L.. Whaley. K. B.. Physical Review Letters. 81. 12. 2594–2597. 13979882.

- Beau . M. . Kiukas . J. . Egusquiza . I. L. . del Campo . A. . Nonexponential quantum decay under environmental decoherence . Phys. Rev. Lett. . 119 . 130401 . 2017 . 13 . 10.1103/PhysRevLett.119.130401 . 29341721 . 2017PhRvL.119m0401B . 1706.06943 . 206299205 .
- Xu . Z. . García-Pintos . L. P. . Chenu . A. . del Campo . A. . Extreme Decoherence and Quantum Chaos . Phys. Rev. Lett. . 122 . 014103 . 2019 . 1 . 10.1103/PhysRevLett.122.014103 . 31012673 . 2019PhRvL.122a4103X . 1810.02319 . 53628496 .
- Web site: Quantum Decoherence and the Measurement Problem. Dan Stahlke. 23 July 2011.
- M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. M. Raimond, S. Haroche . Observing the Progressive Decoherence of the "Meter" in a Quantum Measurement . Phys. Rev. Lett. . 77 . 24 . 4887–4890 . 9 December 1996 . 10.1103/PhysRevLett.77.4887 . 1996PhRvL..77.4887B . 10062660. free .
- Web site: Discovery may overcome obstacle for quantum computing: UBC, California researchers. University of British Columbia. Our theory also predicted that we could suppress the decoherence, and push the decoherence rate in the experiment to levels far below the threshold necessary for quantum information processing, by applying high magnetic fields. (...)Magnetic molecules now suddenly appear to have serious potential as candidates for quantum computing hardware", said Susumu Takahashi, assistant professor of chemistry and physics at the University of Southern California. "This opens up a whole new area of experimental investigation with sizeable potential in applications, as well as for fundamental work".. 20 July 2011. 23 July 2011.
- Web site: USC Scientists Contribute to a Breakthrough in Quantum Computing. University of California, Santa Barbara. 20 July 2011. 23 July 2011.
- News: Breakthrough removes major hurdle for quantum computing. ZDNet. 20 July 2011. 23 July 2011.
- News: Quantum computers may be destroyed by high-energy particles from space . 7 September 2020 . New Scientist.
- News: Cosmic rays may soon stymie quantum computing . 7 September 2020 . phys.org . en.
- Vepsäläinen . Antti P. . Karamlou . Amir H. . Orrell . John L. . Dogra . Akshunna S. . Loer . Ben . Vasconcelos . Francisca . Kim . David K. . Melville . Alexander J. . Niedzielski . Bethany M. . Yoder . Jonilyn L. . Gustavsson . Simon . Formaggio . Joseph A. . VanDevender . Brent A. . Oliver . William D. . Impact of ionizing radiation on superconducting qubit coherence . Nature . August 2020 . 584 . 7822 . 551–556 . 10.1038/s41586-020-2619-8 . 32848227 . 2001.09190 . 210920566 . 7 September 2020 . en . 1476-4687.
- 10.1238/Physica.Topical.102a00069. Probing quantum mechanics towards the everyday world: where do we stand. 2001. Leggett. A. J.. Physica Scripta. 102. 01. 69-73.
- 10.1088/0953-8984/14/15/201. Testing the limits of quantum mechanics: Motivation, state of play, prospects. 2002. Leggett. A. J.. Journal of Physics: Condensed Matter. 14. 15. R415–R451.
- [David Bohm]
- [David Bohm]
- [Hugh Everett]
- [Wojciech H. Zurek]
- [Wojciech H. Zurek]
- [Roger Penrose]
- [Huw Price]