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| In [[physics]], the CHSH inequality can be used in the proof of [[Bell's theorem]], which states that certain consequences of [[quantum entanglement|entanglement]] in [[quantum mechanics]] cannot be reproduced by [[local hidden variable theory|local hidden variable theories]]. Experimental verification of violation of the inequalities is seen as experimental confirmation that nature cannot be described by local hidden variables theories. CHSH stands for [[John Clauser|John '''C'''lauser]], [[Michael Horne|Michael '''H'''orne]], [[Abner Shimony|Abner '''S'''himony]] and [[Richard Holt (physicist)|Richard '''H'''olt]], who described it in a much-cited paper published in 1969 (Clauser ''et al.'', 1969).<ref name="Clauser-1969"/> They derived the '''CHSH inequality''', which, as with [[John Stewart Bell|John Bell's]] original inequality (Bell, 1964),<ref>{{citation |author=J.S. Bell |title= |year=1964 |journal=Physics |volume=1 |pages=195–200}}, reproduced as Ch. 2 of {{citation |author=J. S. Bell |year=1987 |title=Speakable and Unspeakable in Quantum Mechanics |publisher=Cambridge University Press}}</ref> is a constraint on the statistics of "coincidences" in a [[Bell test experiments|Bell test experiment]] which is necessarily true if there exist underlying local hidden variables ([[local realism]]). This constraint can, on the other hand, be infringed by quantum mechanics.
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| == Statement of the inequality ==
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| The usual form of the CHSH inequality is:
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| (1) {{spaces|6}} − 2 ≤ S ≤ 2,
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| :where
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| (2) {{spaces|6}} ''S'' = ''E''(''a'', ''b'') − ''E''(''a'', ''b''′) + ''E''(''a''′, b) + ''E''(''a''′ ''b''′).
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| ''a'' and ''a''′ are detector settings on side A, ''b'' and ''b''′ on side B, the four combinations being tested in separate subexperiments. The terms ''E''(''a'', ''b'') etc. are the [[quantum correlation]]s of the particle pairs, where the quantum correlation is defined to be the expectation value of the product of the "outcomes" of the experiment, i.e. the statistical average of ''A''(''a'')·''B''(''b''), where ''A'' and ''B'' are the separate outcomes, using the coding +1 for the '+' channel and −1 for the '−' channel. Clauser et al.'s 1969<ref name="Clauser-1969">{{citation |author=J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt |year=1969 |title=Proposed experiment to test local hidden-variable theories |journal=Phys. Rev. Lett. |volume=23 |issue=15 |pages=880–4 |doi=10.1103/PhysRevLett.23.880 |bibcode=1969PhRvL..23..880C}}</ref> derivation was oriented towards the use of "two-channel" detectors, and indeed it is for these that it is generally used, but under their method the only possible outcomes were +1 and −1. In order to adapt it to real situations, which at the time meant the use of polarised light and single-channel polarisers, they had to interpret '−' as meaning "non-detection in the '+' channel", i.e. either '−' or nothing. They did not in the original article discuss how the two-channel inequality could be applied in real experiments with real imperfect detectors, though it was later proved (Bell, 1971)<ref name="Bell-1971">J. S. Bell, in ''Foundations of Quantum Mechanics'', Proceedings of the International School of Physics “Enrico Fermi”, Course XLIX, B. d’Espagnat (Ed.) (Academic, New York, 1971), p. 171 and Appendix B. Pages 171-81 are reproduced as Ch. 4 of J. S. Bell, ''Speakable and Unspeakable in Quantum Mechanics'' (Cambridge University Press 1987)</ref> that the inequality itself was equally valid. The occurrence of zero outcomes, though, means it is no longer so obvious how the values of ''E'' are to be estimated from the experimental data.
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| The mathematical formalism of quantum mechanics predicts a maximum value for S of 2{{sqrt|2}}, which is greater than 2, and CHSH violations are therefore predicted by the theory of quantum mechanics.
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| === A typical CHSH experiment ===
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| [[Image:Two_channel_bell_test.svg|300px|thumb|right|'''Schematic of a "two-channel" Bell test'''<br/>The source S produces pairs of photons, sent in opposite directions. Each photon encounters a two-channel polariser whose orientation can be set by the experimenter. Emerging signals from each channel are detected and coincidences counted by the coincidence monitor CM.]]
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| In practice most actual experiments have used light rather than the electrons that Bell originally had in mind. The property of interest is, in the best known experiments ([[Alain Aspect|Aspect]], 1981-2),<ref name="Aspect-1981">{{citation |author=Alain Aspect, Philippe Grangier, Gérard Roger |year=1981 |title=Experimental Tests of Realistic Local Theories via Bell's Theorem |journal=Phys. Rev. Lett. |volume=47 |issue=7 |pages=460–3 |doi=10.1103/PhysRevLett.47.460 |bibcode=1981PhRvL..47..460A}}</ref><ref name="Aspect-1982a">{{citation |author=Alain Aspect, Philippe Grangier, Gérard Roger |year=1982 |title=Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities |journal=Phys. Rev. Lett. |volume=49 |issue=2 |page=91 |doi=10.1103/PhysRevLett.49.91 |bibcode=1982PhRvL..49...91A}}</ref><ref name="Aspect-1982b">{{citation |author=Alain Aspect, Jean Dalibard, Gérard Roger |year=1982 |title=Experimental Test of Bell's Inequalities Using Time-Varying Analyzers |journal=Phys. Rev. Lett. |volume=49 |issue=25 |pages=1804–7 |doi=10.1103/PhysRevLett.49.1804 |bibcode=1982PhRvL..49.1804A}}</ref> the polarisation direction, though other properties can be used. The diagram shows a typical optical experiment. Coincidences (simultaneous detections) are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated.
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| Four separate subexperiments are conducted, corresponding to the four terms ''E''(''a'', ''b'') in the test statistic ''S'' ((2) above). The settings ''a'', ''a''′, ''b'' and ''b''′ are generally in practice chosen to be 0, 45°, 22.5° and 67.5° respectively — the "Bell test angles" — these being the ones for which the QM formula gives the greatest violation of the inequality.
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| For each selected value of ''a'' and ''b'', the numbers of coincidences in each category (''N''<sub>++</sub>, ''N''<sub>--</sub>, ''N''<sub>+-</sub> and ''N''<sub>-+</sub>) are recorded. The experimental estimate for ''E''(''a'', ''b'') is then calculated as:
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| "<math>(3) E = \frac {N_{++} - N_{+-} - N_{-+}+ N_{--}} {N_{++} + N_{+-} + N_{-+}+ N_{--}}</math>
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| Once all the ''E''’s have been estimated, an experimental estimate of ''S'' (expression (2)) can be found. If it is numerically greater than 2 it has infringed the CHSH inequality and the experiment is declared to have supported the QM ([[Quantum Mechanics]]) prediction and ruled out all local hidden variable theories.
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| The CHSH paper lists many preconditions (or "reasonable and/or presumable assumptions") to derive the simplified theorem and formula. For example, for the method to be valid, it has to be assumed that the detected pairs are a fair sample of those emitted. In actual experiments, detectors are never 100% efficient, so that only a sample of the emitted pairs are detected. A subtle, related requirement is that the hidden variables do not influence or determine detection probability in a way that would lead to different samples at each arm of the experiment. | |
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| == Derivation of the CHSH inequality ==
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| The original 1969 derivation will not be given here since it is not easy to follow and involves the assumption that the outcomes are all +1 or −1, never zero. Bell's 1971 derivation is more general. He effectively assumes the "Objective Local Theory" later used by Clauser and Horne (Clauser, 1974).<ref name="Clauser-1974"/> It is assumed that any hidden variables associated with the detectors themselves are independent on the two sides and can be averaged out from the start. Another derivation of interest is given in Clauser and Horne's 1974 paper, in which they start from the CH74 inequality.
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| It would appear from both these later derivations that the only assumptions really needed for the inequality itself (as opposed to the method of estimation of the test statistic) are that the distribution of the possible states of the source remains constant and the detectors on the two sides act independently.
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| === Bell's 1971 derivation ===
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| The following is based on page 37 of Bell's ''Speakable and Unspeakable'' (Bell, 1971),<ref name="Bell-1971"/> the main change being to use the symbol ‘''E''’ instead of ‘''P''’ for the expected value of the quantum correlation. This avoids any implication that the [[quantum correlation]] is itself a probability.
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| We start with the standard assumption of independence of the two sides, enabling us to obtain the joint probabilities of pairs of outcomes by multiplying the separate probabilities, for any selected value of the "hidden variable" λ. λ is assumed to be drawn from a fixed distribution of possible states of the source, the probability of the source being in the state λ for any particular trial being given by the density function ρ(λ), the integral of which over the complete hidden variable space is 1. We thus assume we can write:
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| :<math>E(a, b) = \int \underline {A}(a, \lambda)\underline {B}(b, \lambda)\rho(\lambda)d\lambda \qquad (4)</math>
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| where <u>''A''</u> and <u>''B''</u> are the average values of the outcomes. Since the possible values of ''A'' and ''B'' are −1, 0 and +1, it follows that:
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| :<math>|\underline {A}| \leq 1, |\underline {B}| \leq 1 \qquad (5)</math>
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| Then, if ''a'', ''a''′, ''b'' and ''b''′ are alternative settings for the detectors,
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| :<math>E(a, b) - E(a, b^\prime) = \int [\underline {A}(a, \lambda)\underline {B}(b, \lambda) - \underline {A}(a, \lambda)\underline {B}(b^\prime, \lambda)]\rho(\lambda)d\lambda</math>
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| :<math>= \int \underline {A}(a, \lambda)\underline {B}(b, \lambda)[1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b^\prime, \lambda)]\rho(\lambda)d\lambda</math>
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| :<math>- \int \underline {A}(a, \lambda)\underline {B}(b^\prime, \lambda)[1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b, \lambda)]\rho(\lambda)d\lambda \qquad (6)</math>
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| Then, applying the [[triangle inequality]] to both sides, using (5) and the fact that <math> [1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b^\prime, \lambda)]\rho(\lambda) </math> as well as <math> [1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b, \lambda)]\rho(\lambda) </math> are non-negative we obtain
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| :<math>|E(a, b) - E(a, b^\prime)| \leq \int [1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b^\prime, \lambda)]\rho(\lambda)d\lambda</math>
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| :<math>+ \int [1 \pm \underline {A}(a^\prime, \lambda)\underline {B}(b, \lambda)]\rho(\lambda)d\lambda,</math>
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| or, using the fact that the integral of ρ(λ) is 1,
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| :<math>|E(a, b) - E(a, b^\prime)| \leq 2 \pm [E(a^\prime, b^\prime) + E(a^\prime, b)],</math>
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| which includes the CHSH inequality.
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| === Derivation from Clauser and Horne's 1974 inequality ===
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| In their 1974 paper,<ref name="Clauser-1974">{{citation |author=J.F. Clauser, M.A. Horne |year=1974 |title=Experimental consequences of objective local theories |journal=Phys. Rev. D |volume=10 |issue=2 |pages=526–35 |doi=10.1103/PhysRevD.10.526 |bibcode=1974PhRvD..10..526C}}</ref> Clauser and Horne show that the CHSH inequality can be derived from the CH74 one. As they tell us, in a two-channel experiment the CH74 single-channel test is still applicable and provides four sets of inequalities governing the probabilities ''p'' of coincidences.
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| Working from the inhomogeneous version of the inequality, we can write:
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| (7) {{spaces|6}} −1 ≤ ''p''<sub>''jk''</sub>(''a'', ''b'') − ''p''<sub>''jk''</sub>(''a'', ''b''′) + ''p''<sub>''jk''</sub>(''a''′, ''b'') + ''p''<sub>''jk''</sub>(''a''′, ''b''′) − ''p''<sub>''jk''</sub>(''a''′) − ''p''<sub>''jk''</sub>(''b'') ≤ 0,
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| where ''j'' and ''k'' are each '+' or '−', indicating which detectors are being considered. | |
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| To obtain the CHSH test statistic ''S'' (expression (2)), all that is needed is to multiply the inequalities for which ''j'' is different from ''k'' by −1 and add these to the inequalities for which ''j'' and ''k'' are the same.
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| == Experiments using the CHSH test ==
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| Many Bell test experiments conducted subsequent to [[Alain Aspect|Aspect's]] second experiment in 1982 have used the CHSH inequality, estimating the terms using (3) and assuming fair sampling. Some dramatic violations of the inequality have been reported. (4) Today, this formulation of the Bell inequality remains in use.
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| == See also ==
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| *[[Bell's Theorem]]
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| *[[Bell test experiments]]
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| *[[Leggett–Garg inequality]]
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| *[[Quantum entanglement]]
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| *[[Quantum Mechanics]]
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| == References ==
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| {{reflist}}
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| [[Category:Quantum measurement]]
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| [[Category:Inequalities]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
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