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Quantum entanglement is a Quantum Mechanical phenomenon in which the Quantum State s of two or more Object s have to be described with reference to each other, even though the individual objects may be Spatially Separated . This leads to Correlation s between observable Physical Properties of the System s. For example, it is possible to prepare two particles in a single quantum state such that when one is observed to be Spin -up, the other one will always be observed to be spin-down and vice versa, this despite the fact that it is impossible to Predict , according to quantum mechanics, which set of Measurement s will be observed. As a result, measurements performed on one system seem to be instantaneously influencing other systems entangled with it. But quantum entanglement does ''not'' enable the transmission of classical Information faster than the Speed Of Light (see discussion in next section below). Quantum entanglement has applications in the emerging Technologies of Quantum Computing and Quantum Cryptography , and has been used to realize Quantum Teleportation experimentally. 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, in response to Niels Bohr 's advocacy that quantum mechanics as a theory was complete. Einstein famously derided entanglement as "spukhafte Fernwirkung" or "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, shared, local parameters would cause the correlations. However, in 1964 John Stewart 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. Although there are a number of known loopholes in these experiments, high-efficiency and high-visibility experiments are now in progress which should confirm or disaffirm the influence of those loopholes. For more information, see the article on Bell Test Experiments . Observations on entangled states naively appear to conflict with the property of Relativity that information cannot be transferred faster than the speed of light. Although two entangled systems appear to interact across large spatial separations, no useful information can be transmitted in this way, so Causality cannot be violated through entanglement. This is the statement of No Communication Theorem . 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 required. PURE STATES 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 : | ||
| Not All States Are Product States Fix A | "http://wwwinformationdelightinfo/information/entry/basis_(linear_algebra)" class="copylinks">Basis <math>\{i
angle_A\}</math> for <math>H_A</math> and a basis <math>\{j
angle_B\}</math> for <math>H_B</math> The most general state in <math> H_A \otimes H_B</math> is of the form |
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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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