Difference between revisions 35315396 and 76773363 on enwiki

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, 
:: or else is a member of J and of K, but not of L;
* each member of J which is neither a member of K, nor of L, is therefore a member of J, but not of K; and
* each member of J, which is a member of K, but not of L, is therefore a member of K, but not of L.

The number of members of J which are not members of L is consequently less than, or at most equal to, the sum of the number of members of J which are not members of K, and the number of members of K which are not members of L;

''n<sub>(incl J) (excl L)</sub> &le;≤ n<sub>(incl J) (excl K)</sub> + n<sub>(incl K) (excl L)</sub>''.

If the ratios ''N'' of these numbers to the number ''n<sub>(incl S)</sub>'' of all members of set S can be evaluated, e.g.

''N<sub>(incl J) (excl L)</sub> = n<sub>(incl J) (excl L)</sub> / n<sub>(incl S)</sub>'', 

then the '''Wigner - d'Espagnat inequality''' is obtained as:

''N<sub>(incl J) (excl L)</sub> &le;≤ N<sub>(incl J) (excl K)</sub> + N<sub>(incl K) (excl L)</sub>''.

Considering this particular form in which the Wigner - d'Espagnat inequality is thereby expressed, and noting that the various non-negative ratios ''N'' satisfy 

# ''N<sub>(incl J) (incl K)</sub> + N<sub>(incl J) (excl K)</sub> + N<sub>(excl J) (incl K)</sub> + N<sub>(excl J) (excl K)</sub> = 1'',
(contracted; show full)

'''and''' if A's, B's, and C's channels are considered having been properly ''set up'' only if the constraints <BR>
''orientation angle( A, B ) = orientation angle( B, C ) = orientation angle( A, C )/2 < 
&pi;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> =''<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 &ge;≥ (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>)'',

therefore

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

''(N<sub>(hit A) (miss C)</sub> + N<sub>(miss A) (hit C)</sub>) &ge;≥'' 
::: ''(N<sub>(hit A) (miss B)</sub> + N<sub>(miss A) (hit B)</sub>) + (N<sub>(hit B) (miss C)</sub> + N<sub>(miss B) (hit C)</sub>)'', 

which is in (formal) contradiction to the Wigner - d'Espagnat inequalities

''N<sub>(hit A) (miss C)</sub> &le;≤ N<sub>(hit A) (miss B)</sub> + N<sub>(hit B) (miss C)</sub>'', or <BR>
''N<sub>(miss A) (hit C)</sub>) &le;≤ N<sub>(miss A) (hit B)</sub>) + N<sub>(miss B) (hit C)</sub>)'', or both.

Accordingly, the ratios ''N'' obtained by A, B, and C, with the particular constraints on their ''setup'' in terms of values of orientation angles, '''cannot''' have been derived all at once, in one and the same set of trials together; otherwise they'd necessarily satisfy the Wigner - d'Espagnat inequalities.
(contracted; show full)

==Reference==
* 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]]