That gave me, if I am not mistaken, the last piece of the puzzle. Let’s just take the naive definition of probability—the relative frequency of outcomes as N goes to infinity. Now prepare N systems independently in the state a|0>+b|1>. Now measure one after another—couple the measurement device to the system. At first we have
(a|0>+b|1>)^N |0>.
Now the first one is measured:
(a|0>+b|1>)^(N-1) (a|0,0>+b|1,1>)
where the number after the comma denotes the state of the measuring device, which just counts the number of measured ones. After the second measurement we have
(a|0>+b|1>)^(N-2) (a²|00,0>+ab|01,1>+ab|10,1>+b²|11,2>)
Since the two states ab|01,1> and ab|10,1> are not distinguished by the measurement, the basis should be changed—and this is the crucial point: |01>+|10> has a length of sqrt(2), so if we change the basis to |+>=(|01>+|10>)/sqrt(2), we have
(a|0>+b|1>)^(N-2) (a²|00,0>+absqrt(2)|+,1>+b²|11,2>).
The coefficiants are like in the binomial theorem, but note the sqare root!
Continuing, we will get something similar to a binomial distribution:
Now it remains to prove that for j/N not equal to a² the amplitudes go to zero as N goes to infinity. This is equivalent to the square of the amplitude going to zero (this is just to make the calculation easier, it does not have anything to do with the Born rule). It is, for |...,k>,
ck² = N!/(k!(N-k)!) a²^k b²^(N-k)
which becomes a Gaussian distribution for large N, with mean at k=Na² and width Na²b². So at k/N=a²+d it has a value proportional to exp(-(Nd)²/(2Na²b²))=exp(-Nd²/(2a²b²)) --> 0 as N --> inf.
So a time capsule where the records indicate that some quantum experiment has been performed a great number of times and the Born rule is broken will have an amplitude that goes to zero (yeah, I just read Barbour’s book).
Yes, this is called the Finkelstein-Hartle theorem (D. Finkelstein, Transactions of the New York Academy of Sciences 25, 621 (1963); J. B. Hartle, Am. J. Phys. 36, 704 (1968)).
This theorem is the basis for constructing a limit operator for the relative frequency when there are infinitely many independent repetitions of a measurement, and showing that the product wave-function is an exact eigenstate of the relative frequency operator. Unfortunately, it seems that Hartle’s construction of the frequency operator wasn’t quite right, and needed to be generalized. (E. Farhi, J. Goldstone, and S. Gutmann, Ann. Phys. 192, 368 (1989)).
Even so, the critics are still picky about the construction. There is a line of criticism that infinite frequency operators can be constructed arbitrarily as functions over Hilbert space, and unless you already know the Born rule, you won’t know how to construct one sensibly (so that the Hartle derivation is circular). However this seems unfair, because if you want the relative frequency operator to obey the Kolmogorov axioms of probability then it has to coincide with the Born rule, something which is another long-standing result called Gleason’s theorem. (The squared modulus of the amplitude is the only function of the measure which follows the axioms of probability.) Hence the full derivation is:
1) (Postulate) If the wavefunction is in an eigenstate of a measurement operator, then the measurement will with certainty have the corresponding eigenvalue.
2) (Postulate) Probability is relative frequency over infinitely many independent repetitions.
3) (Postulate) Relative frequency follows the Kolmogorov axioms of probability.
4) (Gleason’s theorem) Relative frequency must converge to the Born rule (squared modulus of amplitude) over infinitely many repetitions, or it won’t be able to follow the Kolmogorov axioms.
5) (Hartle’s theorem, as strengthened by Farhi et al) There is a unique definition of the relative frequency operator over infinite repetitions, and such that the infinite product state is an eigenstate of the relative frequency operator.
6) (Conclusion) The relative frequency over infinitely many measurements is with certainty the Born probability.
That gave me, if I am not mistaken, the last piece of the puzzle. Let’s just take the naive definition of probability—the relative frequency of outcomes as N goes to infinity. Now prepare N systems independently in the state a|0>+b|1>. Now measure one after another—couple the measurement device to the system. At first we have (a|0>+b|1>)^N |0>. Now the first one is measured: (a|0>+b|1>)^(N-1) (a|0,0>+b|1,1>) where the number after the comma denotes the state of the measuring device, which just counts the number of measured ones. After the second measurement we have (a|0>+b|1>)^(N-2) (a²|00,0>+ab|01,1>+ab|10,1>+b²|11,2>) Since the two states ab|01,1> and ab|10,1> are not distinguished by the measurement, the basis should be changed—and this is the crucial point: |01>+|10> has a length of sqrt(2), so if we change the basis to |+>=(|01>+|10>)/sqrt(2), we have (a|0>+b|1>)^(N-2) (a²|00,0>+absqrt(2)|+,1>+b²|11,2>).
The coefficiants are like in the binomial theorem, but note the sqare root!
Continuing, we will get something similar to a binomial distribution:
sum(k=0..N: sqrt(N!/(k!(N-k)!))a^k b^(N-k) |...,k>).
Now it remains to prove that for j/N not equal to a² the amplitudes go to zero as N goes to infinity. This is equivalent to the square of the amplitude going to zero (this is just to make the calculation easier, it does not have anything to do with the Born rule). It is, for |...,k>,
ck² = N!/(k!(N-k)!) a²^k b²^(N-k)
which becomes a Gaussian distribution for large N, with mean at k=Na² and width Na²b². So at k/N=a²+d it has a value proportional to exp(-(Nd)²/(2Na²b²))=exp(-Nd²/(2a²b²)) --> 0 as N --> inf.
So a time capsule where the records indicate that some quantum experiment has been performed a great number of times and the Born rule is broken will have an amplitude that goes to zero (yeah, I just read Barbour’s book).
Yes, this is called the Finkelstein-Hartle theorem (D. Finkelstein, Transactions of the New York Academy of Sciences 25, 621 (1963); J. B. Hartle, Am. J. Phys. 36, 704 (1968)).
This theorem is the basis for constructing a limit operator for the relative frequency when there are infinitely many independent repetitions of a measurement, and showing that the product wave-function is an exact eigenstate of the relative frequency operator. Unfortunately, it seems that Hartle’s construction of the frequency operator wasn’t quite right, and needed to be generalized. (E. Farhi, J. Goldstone, and S. Gutmann, Ann. Phys. 192, 368 (1989)).
Even so, the critics are still picky about the construction. There is a line of criticism that infinite frequency operators can be constructed arbitrarily as functions over Hilbert space, and unless you already know the Born rule, you won’t know how to construct one sensibly (so that the Hartle derivation is circular). However this seems unfair, because if you want the relative frequency operator to obey the Kolmogorov axioms of probability then it has to coincide with the Born rule, something which is another long-standing result called Gleason’s theorem. (The squared modulus of the amplitude is the only function of the measure which follows the axioms of probability.) Hence the full derivation is:
1) (Postulate) If the wavefunction is in an eigenstate of a measurement operator, then the measurement will with certainty have the corresponding eigenvalue.
2) (Postulate) Probability is relative frequency over infinitely many independent repetitions.
3) (Postulate) Relative frequency follows the Kolmogorov axioms of probability.
4) (Gleason’s theorem) Relative frequency must converge to the Born rule (squared modulus of amplitude) over infinitely many repetitions, or it won’t be able to follow the Kolmogorov axioms.
5) (Hartle’s theorem, as strengthened by Farhi et al) There is a unique definition of the relative frequency operator over infinite repetitions, and such that the infinite product state is an eigenstate of the relative frequency operator.
6) (Conclusion) The relative frequency over infinitely many measurements is with certainty the Born probability.
It seem pretty clean to me.