"Coincidence counting" is the experimental technique for the measurement of the observable that says "Thing A happens AND thing B happens". It is not necessarily related to quantum mechanics (although in the case of entangled states it takes on a very essential meaning).
When assessing a probabilistic statement, such as "I throw heads", experimentally, one needs (in the frequentist interpretation, the only one that makes sense to an experimentalist !) to count the number of times that the statement is verified. When the probabilistic statement is a compound statement such as "Joe throws heads, and Jack throws tails" (of course for the *same* event), one can do two things: register independently what Joe throws, and what Jack throws, record all the data (and a means of knowing when they are supposed to be belonging to the same event, like a time stamp)... OR one can make an electronic circuit which only counts when the specific condition is satisfied, in this case the SIMULTANEOUS occurence of "Joe throws heads" and "Jack throws tails". In that case, it is not necessary to record all those data, you just count the number of times that the electronic circuit registered the right condition. This is a very laborious description of the use of an AND gate, of course.
The probability we are assessing is essentially P(A sect B) where A is the event "Joe throws heads" and B is the event "jack throws tails" (if we have another means of determining the total number of events). This is the way to do things, independent of whether we are doing a "quantum experiment" or a "classical experiment" to find the JOINT probability of events A and B.
Quantum mechanically, the eigenstate that corresponds to the OBSERVABLE that describes a JOINT event is a product state of the two systems at hand. If S1 is system 1 (with hilbert space H1) and S2 is system 2 with hilbert space H2, then the hilbert space of the total system (S1 and S2) is H1 x H2. Observables that only observe something wrt S1 are of the form O1 x 1 (unity operator acting on H2), and observables that only observe something wrt S2 are of the form 1 x O2 (unity operator acting on H1).
But observables related to a joint event act on both hilbert spaces, and will have specific eigenvectors of the form |s1> x |s2>. It is the amplitude (squared) of THESE states that we assess when we use coincidence counting, because they determine the JOINT PROBABILITY of having system 1 in state s1 and system 2 in state s2 (in other words, of having the component |s1> x |s2> in the overall system state under observation).
cheers,
Patrick.
:zzz: 
