Editing Quantum decoherence (section)

==Mechanisms==
To examine how decoherence operates, an "intuitive" model is presented below. The model requires some familiarity with quantum theory basics. Analogies are made between visualizable classical [[phase space]]s and [[Hilbert space]]s. 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.

[[File:Quantum superposition of states and decoherence.ogv|thumb|upright=1.5|Quantum superposition of states and decoherence measurement through [[Rabi cycle|Rabi oscillations]]]]

===Phase-space picture===
An ''N''-particle system can be represented in non-relativistic quantum mechanics by a [[wave function]] <math>\psi(x_1, x_2, \dots, x_N)</math>, where each ''x<sub>i</sub>'' 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 6''N'' dimensions (each particle contributes 3 spatial coordinates and 3 momenta). In this case a "quantum" phase space, on the other hand, involves a complex-valued function on a 3''N''-dimensional space. The position and momenta are represented by operators that do not [[Commutativity|commute]], and <math>\psi</math> 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 [[Linear subspace|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 (physics and chemistry)|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 (the environment being one 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. Choosing 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 further interference. The process is effectively [[Reversible process (thermodynamics)|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]].<ref name="zurek03">{{cite journal |last1=Zurek |first1=Wojciech H. |year=2003 |title=Decoherence, einselection, and the quantum origins of the classical |arxiv=quant-ph/0105127 |journal=Reviews of Modern Physics |volume=75 |issue=3| page=715 |doi=10.1103/revmodphys.75.715 |bibcode=2003RvMP...75..715Z |s2cid=14759237}}</ref> 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. This provides one explanation of how the Born rule coefficients effectively act as probabilities as per the measurement postulate constituting a solution to the quantum measurement problem.

===Dirac notation===
Using [[Dirac notation]], let the system initially be in the state

: <math>|\psi\rang = \sum_i |i\rang \lang i |\psi\rang,</math>

where the <math>|i\rang</math>s form an einselected [[eigenbasis|basis]] (''environmentally induced selected eigenbasis''<ref name="zurek03"/>), and let the environment initially be in the state <math>|\epsilon\rang</math>. The [[vector basis]] of the combination of the system and the environment consists of the [[tensor product]]s of the basis vectors of the two subsystems. Thus, before any interaction between the two subsystems, the joint state can be written as

: <math>|\text{before}\rang = \sum_i |i\rang |\epsilon\rang \lang i|\psi\rang,</math>

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

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

: <math>|i\rang |\epsilon\rang </math> evolves into <math>|\epsilon_i\rang,</math>

and so

: <math>|\text{before}\rang</math> evolves into <math>|\text{after}\rang = \sum_i |\epsilon_i\rang \lang i|\psi\rang.</math>

The [[unitarity]] of time evolution demands that the total state basis remains [[orthonormal]], i.e. the [[scalar product|scalar]] or [[inner product]]s of the basis vectors must vanish, since <math>\lang i|j\rang = \delta_{ij}</math>:

: <math>\lang\epsilon_i|\epsilon_j\rang = \delta_{ij}.</math>

This orthonormality of the environment states is the defining characteristic required for einselection.<ref name="zurek03"/>

====System not disturbed by environment====
In an idealized 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

: <math>|i\rang |\epsilon\rang</math> evolves into the product <math>|i, \epsilon_i\rang = |i\rang |\epsilon_i\rang,</math>

and so

: <math>|\text{before}\rang</math> evolves into <math>|\text{after}\rang = \sum_i |i, \epsilon_i\rang \lang i|\psi\rang.</math>

In this case, [[unitarity (physics)|unitarity]] demands that

: <math>\lang i, \epsilon_i|j, \epsilon_j\rang = \lang i|j\rang \lang\epsilon_i|\epsilon_j\rang = \delta_{ij} \lang\epsilon_i|\epsilon_j\rang = \delta_{ij} \lang\epsilon_i|\epsilon_i\rang = \delta_{ij},</math>

where <math>\lang \epsilon_i | \epsilon_i \rang = 1</math> was used. ''Additionally'', decoherence requires, by virtue of the large number of hidden degrees of freedom in the environment, that

: <math>\lang\epsilon_i|\epsilon_j\rang \approx \delta_{ij}.</math>

As before, this is the defining characteristic for decoherence to become einselection.<ref name="zurek03"/> The approximation becomes more exact as the number of environmental degrees of freedom affected increases.

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

===Loss of interference and the transition from quantum to classical probabilities===
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 [[Quantum state|transition]] from <math>\psi</math> to <math>\phi</math> ''before'' <math>\psi</math> has interacted with its environment, then application of the [[Born probability]] rule states that the transition probability is the [[squared modulus]] of the scalar product of the two states:

:<math>\operatorname{prob}_\text{before}(\psi \to \phi) = \left|\lang\psi|\phi\rang\right|^2 = \left|\sum_i \psi^*_i \phi_i\right|^2 = \sum_i |\psi_i^* \phi_i|^2 + \sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i ,</math>

where <math>\psi_i = \lang i|\psi\rang</math>, <math>\psi_i^* = \lang\psi|i\rang</math>, and <math>\phi_i = \lang i|\phi\rang</math> etc.

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

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

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

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

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

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

: <math>\sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i</math>

have vanished from the transition-probability calculation. The decoherence has [[Reversible process (thermodynamics)|irreversibly]] converted quantum behaviour (additive [[probability amplitude]]s) to classical behaviour (additive probabilities).<ref name="zurek03"/><ref name="zurek91">[[Wojciech H. Zurek]], "Decoherence and the transition from quantum to classical", ''Physics Today'', 44, pp. 36–44 (1991).</ref><ref name=Zurek02>{{cite journal
 | last = Zurek
 | first = Wojciech
 | title = Decoherence and the Transition from Quantum to Classical—Revisited
 | journal = Los Alamos Science
 | volume = 27
 | year = 2002
 | url = https://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf
| bibcode = 2003quant.ph..6072Z
 | arxiv = quant-ph/0306072
 }}</ref> 
However, Ballentine<ref>{{Cite journal |last=Ballentine |first=Leslie |date=October 2008 |title=Classicality without Decoherence: A Reply to Schlosshauer |journal=Foundations of Physics |language=en |volume=38 |issue=10 |pages=916–922 |doi=10.1007/s10701-008-9242-0 |issn=0015-9018|doi-access=free |bibcode=2008FoPh...38..916B }}</ref> shows that the significant impact of decoherence to reduce interference need not have significance for the transition of quantum systems to classical limits.

In terms of density matrices, the loss of interference effects corresponds to the diagonalization of the "environmentally traced-over" density matrix.<ref name="zurek03"/>

=== Density-matrix approach ===
The effect of decoherence on [[density matrix|density matrices]] is essentially the decay or rapid vanishing of the [[off-diagonal element]]s of the [[partial trace]] of the joint system's density matrix, i.e. the [[trace (linear algebra)|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"<ref name="zurek03"/> 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.<ref name="zurek03"/> 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

: <math>\rho = |\text{before}\rang \lang\text{before}| = |\psi\rang \lang\psi| \otimes |\epsilon\rang \lang\epsilon|,</math>

where <math>|\epsilon\rang</math> is the state of the environment.
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 [[Quantum entanglement#Reduced density matrices|traced out]], leaving the reduced density matrix for the system:

: <math>\rho_\text{sys} = \operatorname{Tr}_\textrm{env}(\rho) = |\psi\rang \lang\psi| \lang\epsilon|\epsilon\rang = |\psi\rang \lang\psi|.</math>

Now the transition probability will be given as

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

where <math>\psi_i = \lang i|\psi\rang</math>, <math>\psi_i^* = \lang \psi|i\rang</math>, and <math>\phi_i = \lang i|\phi\rang</math> etc.

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

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

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):<ref name="JZ">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.</ref>

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

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

: <math>\sum_j |\phi_j|^2 |j\rang \lang j|.</math>

The transition probability will then be given as

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

which has no contribution from the interference terms

: <math>\sum_{ij; i \ne j} \psi^*_i \psi_j \phi^*_j \phi_i.</math>

The density-matrix approach has been combined with the [[De Broglie–Bohm theory|Bohmian approach]] to yield a ''reduced-trajectory approach'', taking into account the system [[reduced density matrix]] and the influence of the environment.<ref>{{Cite journal |last1=Sanz |first1=A. S. |last2=Borondo |first2=F. |date=2007 |title=A quantum trajectory description of decoherence |journal=The European Physical Journal D |volume=44 |issue=2 |pages=319–326 |doi=10.1140/epjd/e2007-00191-8 |arxiv=quant-ph/0310096 |bibcode=2007EPJD...44..319S |s2cid=18449109 |issn=1434-6060}}</ref>

===Operator-sum representation===
Consider a system ''S'' and environment (bath) ''B'', which are closed and can be treated quantum-mechanically. Let <math>\mathcal H_S</math> and <math>\mathcal H_B</math> be the system's and bath's Hilbert spaces respectively. Then the Hamiltonian for the combined system is

: <math>\hat{H} = \hat H_S \otimes \hat I_B + \hat I_S \otimes \hat H_B + \hat H_I,</math>

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

: <math>\rho_{SB}(t) = \hat U(t) \rho_{SB}(0) \hat U^\dagger(t),</math>

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

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

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

: <math>\hat H_I = \sum_i \hat S_i \otimes \hat B_i,</math>

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

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

<math>\rho_S(t)</math> is called the ''reduced density matrix'' and gives information about the system only. If the bath is written in terms of its set of orthogonal basis kets, that is, if it has been initially diagonalized, then <math>\textstyle\rho_B(0) = \sum_j a_j |j\rangle \langle j|</math>. Computing the partial trace with respect to this (computational) basis gives

: <math>\rho_S(t) = \sum_l \hat A_l \rho_S(0) \hat A^\dagger_l,</math>

where <math>\hat A_l, \hat A^\dagger_l</math> are defined as the ''Kraus operators'' and are represented as (the index <math>l</math> combines indices <math>k</math> and <math>j</math>):

: <math>\hat A_l = \sqrt{a_j} \langle k| \hat U |j\rangle.</math>

This is known as the ''[[Decoherence-free subspaces#Operator-sum representation formulation|operator-sum representation]]'' (OSR). A condition on the Kraus operators can be obtained by using the fact that <math>\operatorname{Tr}[\rho_S(t)] = 1</math>; this then gives

: <math>\sum_l \hat A^\dagger_l \hat A_l = \hat I_S.</math>

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

===Semigroup approach===
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]]<ref name=Belavkin89>{{cite journal
 | author = V. P. Belavkin
 | title = A new wave equation for a continuous non-demolition measurement
 | journal = Physics Letters A
 | volume = 140
 | number = 7–8
 | pages = 355–358
 | year = 1989
 | doi = 10.1016/0375-9601(89)90066-2
 | arxiv = quant-ph/0512136|bibcode = 1989PhLA..140..355B | s2cid = 6083856
 }}</ref><ref name=Carmichael93>{{cite book
 | author = Howard J. Carmichael
 | title = An Open Systems Approach to Quantum Optics
 | publisher = Springer-Verlag
 | year = 1993
 | location = Berlin Heidelberg New-York}}</ref><ref name=Bauer2012>{{cite tech report |author1=Michel Bauer |author2=Denis Bernard |author3=Tristan Benoist | title = Iterated Stochastic Measurements | arxiv = 1210.0425|bibcode=2012JPhA...45W4020B|doi=10.1088/1751-8113/45/49/494020}}</ref> for the evolution under continuous measurement). This uses the [[Quantum states#Schrödinger picture vs. Heisenberg picture|Schrödinger]] picture, where evolution of the ''state'' (represented by its density matrix) is considered. The master equation is

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

where <math>\tilde H_S = H_S + \Delta</math> is the system Hamiltonian <math>H_S</math> along with a (possible) unitary contribution <math>\Delta</math> from the bath, and <math>L_D</math> is the ''Lindblad decohering term''.<ref name="Lidar and Whaley"/> The [[Lindblad equation|Lindblad decohering term]] is represented as
: <math>L_D\big[\rho_S(t)\big] = \frac{1}{2} \sum_{\alpha, \beta = 1}^M b_{\alpha\beta} \Big(\big[\mathbf F_\alpha, \rho_S(t)\mathbf F^\dagger_\beta\big] + \big[\mathbf F_\alpha \rho_S(t), \mathbf F^\dagger_\beta\big]\Big).</math>

The <math>\{\mathbf{F}_\alpha\}_{\alpha=1}^M</math> are basis operators for the ''M''-dimensional space of [[bounded operator]]s that act on the system Hilbert space <math>\mathcal H_S</math> and are the ''error generators''.<ref name="Lidar, Chuang, and Whaley">* {{cite journal |arxiv=quant-ph/9807004 |doi=10.1103/PhysRevLett.81.2594 |bibcode=1998PhRvL..81.2594L |title=Decoherence-Free Subspaces for Quantum Computation |year=1998 |last1=Lidar |first1=D. A. |last2=Chuang |first2=I. L. |last3=Whaley |first3=K. B. |journal=Physical Review Letters |volume=81 |issue=12 |pages=2594–2597 |s2cid=13979882}}</ref> The matrix elements <math>b_{\alpha\beta}</math> represent the elements of a [[Positive semidefinite matrix|positive semi-definite]] [[Hermitian matrix]]; they characterize the decohering processes and, as such, are called the ''noise parameters''.<ref name="Lidar, Chuang, and Whaley"/> The semigroup approach is particularly nice, because it distinguishes between the unitary and decohering (non-unitary) processes, which is not the case with the OSR. In particular, the non-unitary dynamics are represented by <math>L_D</math>, whereas the unitary dynamics of the state are represented by the usual [[Heisenberg commutator]]. Note that when <math>L_D\big[\rho_S(t)\big] = 0</math>, the dynamical evolution of the system is unitary. The conditions for the evolution of the system density matrix to be described by the master equation are:<ref name="Lidar and Whaley"/>
# 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