9 Beyond “either-​​or”

Sup­pose that the sub­spaces A and B rep­re­sent two pos­sible out­comes of the same mea­sure­ment. What mea­sure­ment out­come is rep­re­sented by the span AUB of A and B? (You will remember that the span of A and B is the smallest sub­space con­taining both A and B.)

The fol­lowing obser­va­tions are rel­e­vant here. Let p(A) and p(B) be the respec­tive prob­a­bil­i­ties of obtaining the out­comes rep­re­sented by A and B. Because a line that is con­tained in either A or B is con­tained in AUB, we have that

p(AUB) = 1   when­ever   [p(A) = 1 or p(B) = 1].

Because a line orthog­onal to both A and B is orthog­onal to AUB, we also have that

p(AUB) = 0   when­ever   [p(A) = 0 and p(B) = 0].

This holds, in par­tic­ular, if A and B rep­re­sent dis­joint (non-​​overlapping) inter­vals A and B in the range of a con­tin­uous vari­able Q. What is impor­tant here is that a line can be in AUB without being con­tained in either A or B. This means that the out­come AUB can be cer­tain even if nei­ther A nor B is cer­tain. Obtaining the out­come AUB there­fore does not imply that the value of Q is either A or B, let alone a def­i­nite number in either A or B.

Imagine two per­fect — one hun­dred per­cent effi­cient — detec­tors D(A) and D(B) mon­i­toring the two inter­vals. If the prob­a­bil­i­ties p(A) and p(B) are both greater than 0 (and there­fore less than 1), then it isn’t cer­tain that D(A) will click, and it isn’t cer­tain that D(B) will click. Yet if p(AUB) = 1, then it is cer­tain that either D(A) or D(B) will click. How come? What makes this certain?

The answer lies in the fact that quantum-​​mechanical prob­a­bility assign­ments are invari­ably made on the (tacit) assump­tion that a mea­sure­ment is suc­cess­fully per­formed; there is an out­come. For instance, if A and B are dis­joint regions of space, and if a mea­sure­ment has indi­cated the pres­ence of a par­ticle in the union of these regions, then the tacit assump­tion is that a sub­se­quent posi­tion mea­sure­ment made with two detec­tors mon­i­toring A and B, respec­tively, will yield an out­come — either of the detec­tors will click.

So there is no mys­tery here, but the impli­ca­tion is that quantum mechanics only gives us prob­a­bil­i­ties with which this or that out­come is obtained in a suc­cessful mea­sure­ment. It does not give us the prob­a­bility with which a prop­erty or value is pos­sessed, inde­pen­dently from mea­sure­ments, nor does it allow us to cal­cu­late the prob­a­bility with which an attempted mea­sure­ment will suc­ceed. Hence it is inca­pable of for­mu­lating suf­fi­cient con­di­tions for the suc­cess of a measurement.

Now con­sider the fol­lowing two mea­sure­ments. The first, M1, has three pos­sible out­comes: A, B, and C. The second, M2, has two: AUB and C. We there­fore have

p(A) + p(B) + p(C) = 1   as well as   p(AUB) + p(C) = 1.

It seems rea­son­able to assume non-​​contextuality, which means that it does not make a dif­fer­ence what the other pos­sible out­comes (besides C) are — only AUB or A and B. If this assump­tion is cor­rect, then p(C) is the same in both equa­tions, and

p(AUB) = p(A) + p(B).

Let us remember our objec­tive. We are looking for a prob­a­bility algo­rithm that is capable of accom­mo­dating non­trivial prob­a­bil­i­ties and incom­pat­ible ele­men­tary tests. If common sense in the form of non-​​contextuality is con­sis­tent with these require­ments, we go for it. There is no need to make the world stranger than it already is. By hind­sight we know that Nature concurs.

How­ever, we also know that con­tex­tu­ality is an inescapable fea­ture of sit­u­a­tions in which prob­a­bil­i­ties are assigned either on the basis of past and future out­comes or to out­comes of mea­sure­ments per­formed on entan­gled sys­tems. We encoun­tered two exam­ples of such sys­tems in Sec­tion 1.1 and 1.7. (This con­tex­tu­ality, which per­tains to actual mea­sure­ment con­texts, must be dis­tin­guished from that which the pro­po­nents of pre-​​existent values have in mind.)