Quantum decoherence Totally Explained

Everything about Decoherence totally explained

In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior—a feature of classical physics—and give the appearance of wave function collapse. Decoherence occurs when a system interacts with its environment, or any complex external system, in a thermodynamically irreversible way that ensures different elements in the quantum superposition of the system+environment's wavefunction can no longer interfere with each other. Decoherence has been a subject of active research for the last two decades (External Link

).
Decoherence doesn't provide a mechanism for an actual wave function collapse; rather it provides a mechanism for the appearance of wavefunction collapse. The quantum nature of the system is simply "leaked" into the environment so that a total superposition of the wavefunction still exists, but exists beyond the realm of measurement.
Decoherence represents a major problem for the practical realization of quantum computers, since these heavily rely on the undisturbed evolution of quantum coherences.

Mechanisms

Decoherence isn't a new theoretical framework, but instead a set of new theoretical perspectives in which the environment is no longer ignored in modeling systems. To examine how decoherence operates we'll present an "intuitive" model (which does require some familiarity with the basics of quantum theory) making analogies between visualisable classical phase spaces and Hilbert spaces before presenting a more rigorous derivation of how decoherence destroys interference effects and the "quantum nature" of systems, in Dirac notation. Then the density matrix approach will be presented for perspective (there are many different ways of understanding decoherence).

Phase space picture

An N-particle system can be represented in non-relativistic quantum mechanics by a wavefunction,

psi(x_1, x_2, ...,x_N)

, which 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), whereas our "quantum" phase space contains a complex-valued function in a 3N dimensional space (since the position and momenta don't commute) but can still inherit much of the mathematical structure of a Hilbert space. Aside from these differences, however, the analogy holds.
   Different previously isolated, non-interacting systems occupy different phase spaces, or alternatively we can say 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 3 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, but 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. (Think, by analogy, of a square (2-d surface) extended by just one dimension (a line) to form a cube. The cube has a greater volume, in some sense, than its component square and line axes.) The relevance of this is that the extent that two vectors interfere with each other is a measure of how "close" they're to each other (formally, their overlap or Hilbert space scalar product 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 wavefunction can be expanded as a sum of elements in a quantum superposition, in a quite arbitrary way. Each expansion corresponds to a projection of the wave vector onto a basis, and the bases can be chosen at will. Let us choose any expansion where the resulting 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—so much in fact that after a very short interaction there's almost no chance of any further interference and the process is effectively irreversible; the different elements effectively become "lost" from each other in the expanded phase space created by the coupling with the environment. The elements of the original system 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 might be seen in a double-slit experiment. Any elements that decohere from each other via environmental interactions are said to be quantum entangled with the environment. (Note the converse isn't true: not all entangled states are decohered from each other.)
   Any measuring device, in this model, acts as an environment since any measuring device or apparatus, at some stage along the measuring chain, 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 wavefunction 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. As we've seen if the measuring device has many degrees of freedom, it's 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.

Dirac notation

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

|psi 
ang

where ť

|psi 
ang = sum_i |i
ang lang i|psi 
ang

where the

|i
ang

s form an einselected basis (environmentally induced selected eigen basis, 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, and not until the 1980s
/90s did decoherent-based explanations of the appearance of wavefunction collapse become popular, with the greater acceptance of the use of reduced density matrices — beyond the realm of measurement. Thus decoherence, as a philosophical interpretation, amounts to something similar to the many-worlds approach.

Further Information

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