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 Bell) 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_{(incl J) (excl L)} ≤ n_{(incl J) (excl K)} + n_{(incl K) (excl L)}.

If the ratios N of these numbers to the number n_{(incl S)} of all members of set S can be evaluated, e.g.

N_{(incl J) (excl L)} = n_{(incl J) (excl L)} / n_{(incl S)},

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

N_{(incl J) (excl L)} ≤ N_{(incl J) (excl K)} + N_{(incl K) (excl L)}.

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

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 (e.g. as (hit A) vs. (miss A), (hit B) vs. (miss B), and (hit C) vs. (miss C), respectively), over several (at least pairwise defined) trials, then non-negative ratios N may be evaluated, appropriately labelled, and found to satisfy

However, if the pairwise orientation angles between these three observers are determined (following the inverse of a quantum-mechanical interpretation of Malus's law) 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( A, B ) = orientation angle( B, C ) = orientation angle( A, C )/2 < π/4
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)

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