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${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
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Quantum decoherence is the loss of quantum coherence. Quantum decoherence has been studied to understand how quantum systems convert to systems which can be explained by classical mechanics. Beginning out of attempts to extend the understanding of quantum mechanics, the theory has developed in several directions and experimental studies have confirmed some of the key issues. Quantum computing relies on quantum coherence and is the primary practical applications of the concept.

Concept

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 or environmental 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 can be viewed as the loss of information from a system into the environment (often modeled as a heat bath),^[1] 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).^[2] 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.

History and interpretation

Relation to interpretation of quantum mechanics

An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum physics might correspond to experienced reality.^[3] Decoherence calculations can be done in any interpretation of quantum mechanics, since those calculations are an application of the standard mathematical tools of quantum theory. However, the subject of decoherence has been closely related to the problem of interpretation throughout its history.^[4][5]

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, observation indicates that this mixture looks like a proper quantum ensemble in a measurement situation, as the measurements lead to the "realization" of precisely one state in the "ensemble".

The philosophical views of Werner Heisenberg and Niels Bohr have often been grouped together as the "Copenhagen interpretation", despite significant divergences between them on important points.^[6][7] In 1955, Heisenberg suggested that the interaction of a system with its surrounding environment would eliminate quantum interference effects. However, Heisenberg did not provide a detailed account of how this might transpire, nor did he make explicit the importance of entanglement in the process.^[7][8]

Origin of the concepts

Nevill Mott's solution to the iconic Mott problem in 1929 is considered in retrospect to be the first quantum decoherence work.^[9] It was cited by the first modern theoretical treatment.^[10]

Although he did not use the term, the concept of quantum decoherence was first introduced in 1951 by the American physicist David Bohm,^[11][12] who called it the "destruction of interference in the process of measurement". Bohm later used decoherence to handle the measurement process in the de Broglie-Bohm interpretation of quantum theory.^[13]

The significance of decoherence was further highlighted in 1970 by the German physicist H. Dieter Zeh,^[14] and it has been a subject of active research since the 1980s.^[15] 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.^[16]

The study of decoherence as a proper subject began in 1970, with H. Dieter Zeh's paper "On the Interpretation of Measurement in Quantum Theory".^[4][14] Zeh regarded the wavefunction as a physical entity, rather than a calculational device or a compendium of statistical information (as is typical for Copenhagen-type interpretations), and he proposed that it should evolve unitarily, in accord with the Schrödinger equation, at all times. Zeh was initially unaware of Hugh Everett III's earlier work,^[17] which also proposed a universal wavefunction evolving unitarily; he revised his paper to reference Everett after learning of Everett's "relative-state interpretation" through an article by Bryce DeWitt.^[4] (DeWitt was the one who termed Everett's proposal the many-worlds interpretation, by which name it is commonly known.) For Zeh, the question of how to interpret quantum mechanics was of key importance, and an interpretation along the lines of Everett's was the most natural. Partly because of a general disinterest among physicists for interpretational questions, Zeh's work remained comparatively neglected until the early 1980s, when two papers by Wojciech Zurek^[18][19] invigorated the subject. Unlike Zeh's publications, Zurek's articles were fairly agnostic about interpretation, focusing instead on specific problems of density-matrix dynamics. Zurek's interest in decoherence stemmed from furthering Bohr's analysis of the double-slit experiment in his reply to the Einstein–Podolsky–Rosen paradox, work he had undertaken with Bill Wootters,^[20] and he has since argued that decoherence brings a kind of rapprochement between Everettian and Copenhagen-type views.^[4][21]

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—beyond the realm of measurement.^[22][23] 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).

Criticism of the adequacy of decoherence theory to solve the measurement problem has been expressed by Anthony Leggett.^[24][25]

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 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.

Phase-space picture

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 x_i 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). In this case a "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 (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 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.^[26] 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

${\displaystyle |\psi \rangle =\sum _{i}|i\rangle \langle i|\psi \rangle ,}$

where the ${\displaystyle |i\rangle }$s form an einselected basis (environmentally induced selected eigenbasis^[26]), and let the environment initially be in the state ${\displaystyle |\epsilon \rangle }$. 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

${\displaystyle |{\text{before}}\rangle =\sum _{i}|i\rangle |\epsilon \rangle \langle i|\psi \rangle ,}$

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

${\displaystyle |i\rangle |\epsilon \rangle }$ evolves into ${\displaystyle |\epsilon _{i}\rangle ,}$

and so

${\displaystyle |{\text{before}}\rangle }$ evolves into ${\displaystyle |{\text{after}}\rangle =\sum _{i}|\epsilon _{i}\rangle \langle i|\psi \rangle .}$

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 ${\displaystyle \langle i|j\rangle =\delta _{ij}}$:

${\displaystyle \langle \epsilon _{i}|\epsilon _{j}\rangle =\delta _{ij}.}$

This orthonormality of the environment states is the defining characteristic required for einselection.^[26]

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

${\displaystyle |i\rangle |\epsilon \rangle }$ evolves into the product ${\displaystyle |i,{\mathrm{ϵ<!--\; ϵ\; -->}}_i⟩<!--\; ⟩\; -->=|i⟩<!--\; ⟩\; -->|}$Zdroj:https://en.wikipedia.org?pojem=Quantum_decoherence
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