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SUPERPOSITION AND ENTANGLEMENT Decoherence occurs when a system loses phase coherence between different portions of its quantum mechanical state. It then no longer exhibits quantum Interference between those portions (as might be seen in a Double-slit Experiment ). Decoherence is caused by interactions with a second system which may be thought of as either "the environment" or as "a measuring device". In the latter view, 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 portions of the entangled system's state to interfere, the original system and the measuring device must both evolve into the same state. 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 portions rather than as a single coherent Quantum Superposition of them. From the perspective of the measuring device, in each member of the ensemble the system appears to have Collapsed onto a state with precise values for the measured attributes. DECOHERENCE CAN BE RAPID Decoherence represents an extremely fast process for macroscopic objects, since these are interacting with many microscopic objects in their natural environment. The process explains why we tend not to observe quantum behaviour in everyday macroscopic objects despite their existing in a bath of air molecules and photons. It also explains why we do see classical fields from the properties of the interaction between matter and radiation. DECOHERENCE AND MEASUREMENT The discontinuous "wave function collapse" postulated in the Copenhagen Interpretation to enable the theory to be related to the results of laboratory measurements is now to a large extent describable within the normal dynamics of quantum mechanics via the decoherence process. Consequently, decoherence is an important part of the modern version of the Copenhagen interpretation, based on Consistent Histories . Decoherence shows how a macroscopic system interacting with a lot of microscopic systems (e.g. collisions with air molecules or photons) moves from being in a pure quantum state—which in general will be a coherent superposition (see Schrödinger's Cat )—to being in an incoherent mixture of these states. The population of the mixture in case of measurement is exactly that which gives the probabilities of the different results of such a measurement. However, decoherence does not give a complete solution of the Measurement Problem , since all components of the wave function still exist in a global superposition. Decoherence explains why these coherences are no longer available for local observers. MATHEMATICS OF DECOHERENCE Mathematically, the process results in the off diagonal elements of the Density Matrix or state operator of the system vanishing very quickly in a basis, which is usually defined by the interaction Hamiltonian between a system and its environment. Technically, the states of the environment are "averaged over". Decoherence represents a major problem for the practical realization of Quantum Computers , since these heavily rely on undisturbed evolution of quantum coherences. MATHEMATICAL DETAILS Let's assume for the moment the system in question consists of a subsystem being studied, A and the "environment" E, and the total Hilbert Space is the Tensor Product of a Hilbert space describing A, HA and a Hilbert space describing E, HE: that is, :. | ||
| Where &psi<sub>1</sub>> And &psi<sub>2</sub>> Are Orthogonal And There Is No | "http://wwwinformationdelightinfo/encyclopedia/entry/entanglement" class="copylinks">Entanglement initially Also, choose an orthonormal basis for H<sub>A</sub>, |
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| <math> \{ e I Angle \} I</math> (This Could Be A "continuously Indexed Basis" Or A Mixture Of Continuous And Discrete Indexes, In Which Case We Would Have To Use A | "http://wwwinformationdelightinfo/encyclopedia/entry/rigged_Hilbert_space" class="copylinks">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 |
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