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Our example illustrates the distinction between entangled and nonentangled states, but quantum entanglement has the ability to encode oppositeness that goes further than the awake-asleep dichotomy of the cats or the heads-tails dichotomy of coins. One extension is the anti-alignment of the hands of a clock, not uniquely found at 6:00, but realized also at different angles of the hands, at approximately 7:05, 8:10, etc. Orthogonal photon polarizations, which may be oriented to any direction, extend the concept to continuous opposites, exploited by Clauser and Freedman in their famous experiment showing quantum violation of a Bell Inequality 54
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Our example illustrates the distinction between entangled and nonentangled states, but quantum entanglement has the ability to encode oppositeness that goes further than the awake-asleep dichotomy of the cats or the heads-tails dichotomy of coins. One extension is the anti-alignment of the hands of a clock, not uniquely found at 6:00, but realized also at different angles of the hands, at approximately 7:05, 8:10, etc. Orthogonal photon polarizations, which may be oriented to any direction, extend the concept to continuous "opposites," exploited by Clauser and Freedman in their famous experiment showing quantum violation of a Bell Inequality (54).
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Matrix encoding of the entanglement is more complicated than the bracket notation used for the cats, but it is important because we want to allow for more possibilities, which can be visualized by thinking that any of the four joint state labels, or, can be correlated with itself or any other, making 16 basic correlation labels, and these are conventionally arranged in a 4 x 4 matrix. This is the density matrix r referred to in the text. See the Supporting Online Material for an example, and a sketch of the theory of time evolution of the density matrix in a noisy environment
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Matrix encoding of the entanglement is more complicated than the bracket notation used for the cats, but it is important because we want to allow for more possibilities, which can be visualized by thinking that any of the four joint state labels ++, +-, -+, or -- can be correlated with itself or any other, making 16 basic correlation labels, and these are conventionally arranged in a 4 x 4 matrix. This is the density matrix r referred to in the text. See the Supporting Online Material for an example, and a sketch of the theory of time evolution of the density matrix in a noisy environment.
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T.Y. and J.H.E. acknowledge grant support from the U.S. Army Research Office (48422-PH), the NSF (PHY06-01804 and PHY07-58016), and U.S. Department of Energy, Office of Basic Energy Sciences (DE-FG02-05ER15713).
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T.Y. and J.H.E. acknowledge grant support from the U.S. Army Research Office (48422-PH), the NSF (PHY06-01804 and PHY07-58016), and U.S. Department of Energy, Office of Basic Energy Sciences (DE-FG02-05ER15713).
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