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 both employed it in their popularizations of quantum mechanics.
Given a set S with three subsets, J, K, and L, the following holds:
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 ≤ n + n .
If the ratios N of these numbers to the number n of all members of set S can be evaluated, e.g.
N = n / n,
then the Wigner–d'Espagnat inequality is obtained as:
N ≤ N + N .
Considering this particular form in which the Wigner–d'Espagnat inequality is thereby expressed, and noting that the various non-negative ratios N satisfy
  1. N + N + N + N = 1,
  2. N + N + N + N = 1, and
  3. N + N + N + N = 1,
it is probably worth mentioning that certain non-negative ratios are readily encountered, which are appropriately labelled by similarly related indices, and which do satisfy equations corresponding to 1., 2. and 3., but which nevertheless don't satisfy the Wigner–d'Espagnat inequality. For instance:
if three observers, A, B, and C, had each detected signals in one of two distinct own channels vs. ', ' vs. ', and ' vs. , over several trials, then non-negative ratios N may be evaluated, appropriately labelled, and found to satisfy
  1. N + N + N + N = 1,
  2. N + N + N + N = 1, and
  3. N + N + N + N = 1.
However, if the pairwise orientation angles between these three observers are determined from the measured ratios as
and if A's, B's, and C's channels are considered having been properly set up only if the constraints

orientation angle = orientation angle = orientation angle/2 < π/4

had been found satisfied, then necessarily
² =

Since
1 ≥ + N ,
therefore
1 ≥ 2 + N – 1,

+ N – 1) ≥ + N – 1)2,

+ N – 1) ≥ + N ,

+ N ) ≥ + N ),

+ N ≥ 2 + N ,
+ N ≥
which is in contradiction to the Wigner–d'Espagnat inequalities
N ≤ N + N , or

N ≤ N + N , 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.
Instead, they had to be derived in three distinct sets of trials, separately and pairwise by A and B, by A and C, and by B and C, respectively.
The failure of certain measurements to be obtained at once, together from one and the same set of trials, and thus their failure to satisfy Wigner–d'Espagnat inequalities, has been characterized as constituting disproof of Einstein's notion of local realism.
Similar interdependencies between two particular measurements and the corresponding operators are the uncertainty relations as first expressed by Heisenberg for the interdependence between measurements of distance and of momentum, and as generalized by Edward Condon, Howard Percy Robertson, and Erwin Schrödinger.