According to Born's rule, a special case of Gleason's trace rule (page 2 of this article), the probability p(v|w) of obtaining the outcome v right after obtaining the outcome w is

p(v|w) = |<v|w>|².

The validity of this simple formula is restricted to measurements that are repeatable and complete. A measurement is said to be repeatable if the following holds: whenever it is repeated after a negligible time span, it yields the same outcome as before. A measurement is said to be complete if it has the greatest possible number of outcomes.

Starting from Born's rule, let us calculate the probability p(v|w,u) of obtaining the outcome v given that the outcome u is obtained before v, and given that the outcome w is obtained after v. We begin by calculating the probability p(w,v|u) of obtaining first v and then w, given that u was obtained before v. p(w,v|u) is the product of two probabilities: the probability |<v|u>|² of obtaining v given u and the probability |<w|v>|² of obtaining w given v:

p(w,v|u) = |<w|v> <v|u>|².

Next we calculate the probability p_V(w|u) of obtaining w given u, given that a measurement M is made in the meantime, and given that v is a possible outcome of M. p_V(w|u) is obtained by adding the probabilities p(w,v_k|u) of all possible outcomes v_k of M:

p_V(w|u) = Σ_k |<w|v_k> <v_k|u>|².

To obtain p(v|w,u), all we have to do is divide p(w,v|u) by p_V(w|u):

p(v|w,u) = |<w|v> <v|u>|² / Σ_k |<w|v_k><v_k|u>|².

This is the ABL rule, so called because it was first formulated by Aharonov, Bergmann, and Lebowitz.

If the intermediate measurement is not a complete measurement, its outcomes cannot be represented by vectors but must instead be represented by projection operators. If we use capital letters for these projectors, the ABL rule takes the form

p(v|w,u) = |<w|V|u>|² / Σ_k |<w|V_k|u>|².

We are going to apply the ABL rule to the following setup. Instead of a plate with two slits we have a plate with three holes A, B, C. In front of the plate we place our electron gun G, and behind it we place a detector D. Both G and D are equidistant from the holes. Right behind C we put a device that causes a phase shift by π (that is, it has the effect of multiplying the propagator <D|C> by e^iπ = -1).

Experiment illustrating the contextuality of ABL probabilities

Using standard quantum-mechanical mumbo-jumbo, we represent the "prepared" state of the electron by the normalized linear combination of vectors

|prepared> = ( |A> + |B> + |C> ) / √3,

and we represent the detected state by

|detected> = ( |A> + |B> – |C> ) / √3.

(The minus is due to the phase shifting device behind C.) We then insert |u> = |prepared> and |w> = |detected> into the ABL rule and calculate p(v|w,u) for |v> = |A> — that is, the probability with which the electron would have been found going through hole A if the appropriate measurement had been made. Interestingly, this probability is not unique.

If we place right behind A a device that beeps each time an electron goes through this hole, the intermediate measurement has two possible outcomes, "through A" and "not through A", which are represented by the projection operators |A><A| and |B><B| + |C><C|. Plugging these into the ABL rule yields the surprising result that p(v|w,u) = 1. If this intermediate measurement is made (and, not to forget, if the outcomes of the initial and final measurements are as specified), then we are sure to find that the electron went through A.

What is even more shocking is that if the beeper is instead placed behind B, then we are sure to find that the electron went through B! Finally, if there are two beepers, one behind A and one behind B, then the intermediate measurement has three possible outcomes — through A, through B, and through C — and the ABL rule yields probability 1/3 for each.

The bottom line: ABL probabilities are contextual: the probability of a possible outcome (generally) depends on what the other possible outcomes are. (We considered three different measurements each with a different set of possible outcomes.)