13 Whence those “rules of the game”?

Sup­pose that a max­imal test per­formed at the time t1 yields the out­come u, and that we want to cal­cu­late the prob­a­bility with which a max­imal test per­formed at the later time t2 yields the out­come w. Fur­ther sup­pose that at some inter­me­diate time t another max­imal test is made, and that its pos­sible out­comes are v1, v2, v3,… Let v be one of these values. Because a max­imal test ren­ders the out­comes of ear­lier mea­sure­ments irrel­e­vant, the joint prob­a­bility p(w,v|u) with which the inter­me­diate and final tests yield v and w, respec­tively, given the ini­tial out­come u, is the product of two prob­a­bil­i­ties: the prob­a­bility p(v|u) of v given u, and the prob­a­bility p(w|v) of w given v. By Born’s rule, this is

p(w,v|u) = |<w|v> <v|u>|2,

where u, v, and w are unit vec­tors in the sub­spaces rep­re­senting u, v, and w, respec­tively. To obtain the prob­a­bility of w given u, regard­less of the inter­me­diate out­come, we must cal­cu­late this prob­a­bility for all pos­sible inter­me­diate out­comes and add the results:

pA(w|u) = |<w|v1> <v1|u>|2 + |<w|v2> <v2|u>|2 + |<w|v3> <v3|u>|2 + ···

In (other) words, first square the mag­ni­tudes of the ampli­tudes <w|v1> <v1|u>, etc., then add the results. This is Rule A. (Hence the sub­script A.)

Rule B next. Since the vec­tors v1, v2, v3,… form a basis, any vector u can be written as

u = u1v1 + u2v2 + u3v3 + ···,

where u1, u2, u3, … are the com­po­nents of u with respect to this basis. The scalar product of v1 and u is

<v1|u> = u1<v1|v1> + u2<v1|v2> + ···.

Since basis vec­tors are unit vec­tors, we have that <v1|v1> = 1, and since they are mutu­ally orthog­onal, we have that <v1|v2> = 0. Hence <v1|u> = u1, <v2|u> = u2, etc. Consequently,

u = v1 <v1|u> + v2 <v2|u> + v3 <v3|u> + ···

and

<w|u> = <w|v1> <v1|u> + <w|v2> <v2|u> + <w|v3> <v3|u> + ···.

If no inter­me­diate mea­sure­ment is made, the prob­a­bility of w given u is

pB(w|u) = |<w|u>|2 = |<w|v1> <v1|u> + <w|v2> <v2|u> + <w|v3> <v3|u> + ···|2

In (other) words, first add the ampli­tudes <w|v1> <v1|u>, etc., then square the mag­ni­tude of the result. This is Rule B. (Hence the sub­script B.)

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