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Quantum entanglement is closely concerned with the emerging Technologies of Quantum Computing and Quantum Cryptography , and has been used to experimentally realize Quantum Teleportation . At the same time, it prompts some of the more philosophically oriented discussions concerning quantum theory. The correlations predicted by quantum mechanics, and observed in experiment, reject the principle of Local Realism , which is that information about the state of a system should only be mediated by interactions in its immediate surroundings. Different views of what is actually occurring in the process of quantum entanglement can be related to different Interpretations Of Quantum Mechanics . BACKGROUND Entanglement is one of the properties of quantum mechanics which caused Einstein and others to dislike the theory. In 1935, Einstein, Podolsky , and Rosen formulated the EPR Paradox , a quantum-mechanical thought experiment with a highly counterintuitive and apparently nonlocal outcome. Einstein famously derided entanglement as "spooky Action At A Distance ." On the other hand, quantum mechanics has been highly successful in producing correct experimental predictions, and the strong correlations associated with the phenomenon of quantum entanglement have in fact been observed. One apparent way to explain quantum entanglement is an approach known as " Hidden Variable Theory ", in which unknown deterministic microscopic parameters would cause the correlations. However, in 1964 Bell derived an upper limit, known as Bell's Inequality , on the strength of correlations for any theory obeying "local realism" (see Principle Of Locality ). Quantum entanglement can lead to stronger correlations that violate this limit, so that quantum entanglement is experimentally distinguishable from a broad class of local hidden-variable theories. Results of subsequent experiments have overwhelmingly supported quantum mechanics. It is known that there are a number of loopholes in these experiments. High efficiency and high visibility experiments are now in progress which should accept or reject those loopholes. For more information, see the article on Bell Test Experiments . Observations on entangled states naively appear to conflict with the property of Einsteinian results, and (ii) the No Cloning Theorem forbids the statistical inspection of entangled quantum states. Although no information can be transmitted through entanglement alone, it is possible to transmit information using a set of entangled states used in conjunction with a ''classical'' information channel. This process is known as Quantum Teleportation . Despite its name, quantum teleportation cannot be used to transmit information faster than light, because a Classical Information Channel is involved. FORMALISM The following discussion builds on the theoretical framework developed in the articles Bra-ket Notation and Mathematical Formulation Of Quantum Mechanics . Consider two noninteracting systems and , with respective Hilbert Space s and . The Hilbert space of the composite system is the Tensor Product : | ||
| Pick Observables (and Corresponding | "http://wwwinformationdelightinfo/encyclopedia/entry/Hermitian" class="copylinks">Hermitian operators) <math>\Omega_A</math> acting on <math>H_A</math>, and <math>\Omega_B</math> acting on <math>H_B</math> According to the Spectral Theorem , we can find a Basis <math>\{i
angle_A\}</math> for <math>H_A</math> composed of eigenvectors of <math>\Omega_A</math>, and a basis <math>\{j
angle_B\}</math> for <math>H_B</math> composed of eigenvectors of <math>\Omega_B</math> We can then write the above pure state as |
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| <math> Ho | \sum_i w_i \alpha_i
angle \langle\alpha_i</math>, |
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| :<math> Ho T | \Psi
angle \ \langle\Psi</math> |
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| :<math> Ho A \equiv \sum J \langle J B \left( \Psi Angle \langle\Psi Ight) j Angle B | \hbox{Tr}_B \
ho_T </math> |
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| :<math> Ho A | (1/2) \bigg( 0
angle_A \langle 0_A + 1
angle_A \langle 1_A \bigg)</math> |
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| :<math> Ho A | \psi
angle_A \langle\psi_A </math> |
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