Wigner - d'Espagnat inequality

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_{(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

1. N_{(incl J) (incl K)} + N_{(incl J) (excl K)} + N_{(excl J) (incl K)} + N_{(excl J) (excl K)} = 1,
2. N_{(incl J) (incl L)} + N_{(incl J) (excl L)} + N_{(excl J) (incl L)} + N_{(excl J) (excl L)} = 1, and
3. N_{(incl K) (incl L)} + N_{(incl K) (excl L)} + N_{(excl K) (incl L)} + N_{(excl K) (excl L)} = 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 (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

1. N_{(hit A) (hit B)} + N_{(hit A) (miss B)} + N_{(miss A) (hit B)} + N_{(miss A) (miss B)} = 1,
2. N_{(hit A) (hit C)} + N_{(hit A) (miss C)} + N_{(miss A) (hit C)} + N_{(miss A) (miss C)} = 1, and
3. N_{(hit B) (hit C)} + N_{(hit B) (miss C)} + N_{(miss B) (hit C)} + N_{(miss B) (miss C)} = 1.

However if the pairwise orientation angles between these three observers are determined (following Malus's definition) from the measured ratios as

orientation angle( A, B ) = 1/2 arccos( N_{(hit A) (hit B)} - N_{(hit A) (miss B)} - N_{(miss A) (hit B)} + N_{(miss A) (miss B)} ), orientation angle( A, C ) = 1/2 arccos( N_{(hit A) (hit C)} - N_{(hit A) (miss C)} - N_{(miss A) (hit C)} + N_{(miss A) (miss C)} ), orientation angle( B, C ) = 1/2 arccos( N_{(hit B) (hit C)} - N_{(hit B) (miss C)} - N_{(miss B) (hit C)} + N_{(miss B) (miss C)} ),

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)

(cos( orientation angle( A, C ) ))² =

(N_{(hit A) (hit C)} + N_{(miss A) (miss C)}) = (2 (N_{(hit A) (hit B)} + N_{(miss A) (miss B)}) - 1)² > 0.

Since

1 ≥ (N_{(hit A) (hit B)} + N_{(miss A) (miss B)}),

therefore

1 ≥ 2 (N_{(hit A) (hit B)} + N_{(miss A) (miss B)}) - 1,
(2 (N_{(hit A) (hit B)} + N_{(miss A) (miss B)}) - 1) ≥ (2 (N_{(hit A) (hit B)} + N_{(miss A) (miss B)}) - 1) ²,
(2 (N_{(hit A) (hit B)} + N_{(miss A) (miss B)}) - 1) ≥(N_{(hit A) (hit C)} + N_{(miss A) (miss C)}),
(1 - 2 (N_{(hit A) (miss B)} + N_{(miss A) (hit B)})) ≥ (1 - (N_{(hit A) (miss C)} + N_{(miss A) (hit C)})),
(N_{(hit A) (miss C)} + N_{(miss A) (hit C)}) ≥ 2 (N_{(hit A) (miss B)} + N_{(miss A) (hit B)}),

(N_{(hit A) (miss C)} + N_{(miss A) (hit C)}) ≥

(N_{(hit A) (miss B)} + N_{(miss A) (hit B)}) + (N_{(hit B) (miss C)} + N_{(miss B) (hit C)}),

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

N_{(hit A) (miss C)} ≤ N_{(hit A) (miss B)} + N_{(hit B) (miss C)}, or
N_{(miss A) (hit C)}) ≤ N_{(miss A) (hit B)}) + N_{(miss B) (hit C)}), 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 (such as the non-negative ratios in the example) 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.

Reference

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