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Wigner-d'Espagnat inequality
The '''Wigner - d'Espagnat inequality''' is a basic result of [[set theory]].
It is named for [[Eugene Wigner]] and [[Bernard d'Espagnat]] who (as pointed out by [[John Stewart Bell|Bell]]) both employed it in their popularizations of [[quantum mechanics]].

Given a set S with three subsets, J, K, and L, the following holds:

* each member of S which is a member of J, but not of L 
:: is either a member of J, but neither of K, nor of L, 
(contracted; show full)
''orientation angle( A, B ) = orientation angle( B, C ) = orientation angle( A, C )/2 < π/4'' <BR> 
had been found satisfied (as one may well require, to any accuracy; where the accuracy depends on the number of trials from which the orientation angle values were obtained), then necessarily (given sufficient accuracy)

''(cos( orientation angle( A, C ) ))
<sup>2</sup>² =''<BR>
: ''(N<sub>(hit A) (hit C)</sub> + N<sub>(miss A) (miss C)</sub>) = (2 (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>) - 1)<sup>2</sup> > 0''. 

Since

''1 ≥ (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>)'',

therefore

''1 ≥ 2 (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>) - 1'', <BR>
''(2 (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>) - 1) ≥ (2 (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>) - 1) <sup>2</sup>²'', <BR>
''(2 (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>) - 1) ≥(N<sub>(hit A) (hit C)</sub> + N<sub>(miss A) (miss C)</sub>)'', <BR>
''(1 - 2 (N<sub>(hit A) (miss B)</sub> + N<sub>(miss A) (hit B)</sub>)) ≥ (1 - (N<sub>(hit A) (miss C)</sub> + N<sub>(miss A) (hit C)</sub>))'', <BR>
(contracted; show full)

Similar interdependencies between ''two'' particular measurements and the corresponding operators are the [[uncertainty principle|uncertainty relations]] as first expressed by [[Werner Heisenberg|Heisenberg]] for the interdependence between measurements of distance and of momentum, and as generalized by [[Edward Condon]], [[Howard Percy Robertson]], and [[Erwin Schrödinger]].

==Reference
s==
* John S. Bell, ''Bertlmann's socks and the nature of reality'', Journal de Physique '''42''', no. 3, p. 41 (1981); and references therein.

[[Category:Inequalities]]