There have been many questions asked here on the topic of whether the Born rule can be derived from the rest of the axioms of quantum mechanics. See, for example, this and links therein. However, I want to ask about a specific derivation of the Born rule due to Jim Hartle, arXiv:quant-ph/1907.02953v1 which is a $2019$ arXiv repost of his original 1968 paper in American Journal of Physics. One can see this paper referenced, among other places, in Sidney Coleman's famous Dirac lecture originally titled "Quantum Mechanics In Your Face" in the discussion of how probabilities arise in quantum mechanics. Hartle derives the Born rule simply within the standard framework of quantum mechanics by doing something pretty clever but I fail to understand as to exactly how it amounts to deriving the Born rule and if it doesn't then what does it actually amount to. As far as I can see, this question hasn't been discussed here before.
OK, so what does Hartle do?
Hartle considers an ensemble of $N$ identically prepared quantum systems denoted by $\vert \psi^N\rangle\equiv\otimes_{i=1}^{N}\vert\psi\rangle$ which lives in the ensemble Hilbert space constructed by the tensor-product $\mathcal{H}^N$ where $\mathcal{H}$ is the Hilbert space of an individual quantum system, i.e., $\vert \psi\rangle\in\mathcal{H}$.
One can consider an observable $A=\sum_ka_k\vert a_k\rangle\langle a_k\vert$ over $\mathcal{H}$ such that $\vert\psi\rangle$ is not necessarily an eigenstate of $A$.
Now, Hartle constructs a frequency operator $f_k^N$ over the ensemble Hilbert space $\mathcal{H}^N$ corresponding to an eigenvalue $a_k$ of the observable $A$ defined as \begin{align} f_k^N\equiv\sum_{i_1i_2...i_N}\vert a_{i_1}\rangle\otimes\vert a_{i_2}\rangle\otimes...\otimes\vert a_{i_N}\rangle\Big(\frac{1}{N}\sum_{\alpha=1}^{N}\delta_{i_{\alpha}k}\Big)\langle a_{i_N}\vert\otimes...\otimes\langle a_{i_2}\vert\otimes\langle a_{i_1}\vert \end{align} such that it is diagonalized by the eigenstates of $A^N$ and the eigenvalue of $f_k^N$ corresponding to an eigenstate of $A^N$ is the relative frequency of the eigenvalue $a_k$ in this eigenstate, i.e.,
$1/N$ times the number of times $\vert a_k\rangle$ appears in the tensor product created by taking an eigenstate of $\vert A\rangle$ from each of the factors of $\mathcal{H}^N$. I hope that the notational clutter doesn't take away from the conceptual simplicity of the definition of the operator and from the understanding as to why the name "frequency operator" is (at least so far) justified.Now, something dramatic happens in the limit $N\to\infty$. In the limit $N\to\infty$, Hartle shows that all states of the form $\vert \psi^N\rangle$ are eigenstates of this frequency operator with the eigenvalue given by, you guessed it, $\vert\langle\psi\vert a_k\rangle\vert^2$.
OK, all of that is purely deductive, unless one finds a mistake in the math, nothing that one can really object to. Where the meat of my confusion lies is in what Hartle claims he has done by doing the aforementioned calculation.
What does Hartle claim?
- Hartle claims that since a quantity can be said to have a well-defined value for a quantum system if and only if the quantum system is in an eigenstate of the corresponding operator, one ought to say that since $\vert \psi^\infty\rangle$ is an eigenstate of the frequency operator $f_k^{\infty}$, there is a well-defined value for the (relative) frequency of the eigenvalue $a_k$ in this ensemble even if the state $\vert \psi\rangle$ is not an eigenstate of the observable $A$. As already demonstrated, this (relative) frequency is, in particular, given by $\vert\langle\psi\vert a_k\rangle\vert^2$.
- Thus, Hartle claims, what we have shown is that the (relative) frequencies of the results of the measurement of $A$ can be predicted for any given state $\vert\psi\rangle$ and are given by the Born rule.
What I don't understand...
I don't understand how this demonstration says anything at all about what would be the (relative) frequencies of outcomes when we measure $A$ on a state $\vert\psi\rangle$ that is not an eigenstate of $A$. In particular, I think Hartle takes the name "frequency operator" a bit too seriously and attaches to it the meaning of something that gives "the (relative) frequency of getting a certain eigenvalue upon a measurement" whereas its definition simply tells us that it's an operator that gives us "the (relative) frequency of the occurrences of an eigenvalue of $A$ in an ensemble that is already an eigenstate of $A^N$".
Sure, it is interesting that all $\vert\psi^N\rangle$ become eigenstates of this operator in the limit $N\to\infty$ but it doesn't tell us that we have discovered a probability law and rather it just tells us that the operator that we have constructed has peculiar properties in the limit $N\to\infty$. In particular, that in the case of a finite $N$, it is only diagonalized by states who actually have a well-defined number of occurrences of the eigenvalue $a_k$, but, in the $N\to\infty$ it is diagonalized by states who do not have this property. After all, an operator is simply what it has been defined to be, and it is under no obligation to preserve its qualitative behavior during the transition from finite $N$ to the $N\to\infty$ limit.
At a more basic level, if you don't have something equivalent to a collapse postulate of some kind, there is no answer to the question of what happens when you "measure" (which you also have to define somehow) an observable over a state that is not in an eigenstate of the observable. You don't know if the measurement simply doesn't yield anything or if it yields an eigenvalue with some probability or if the whole universe explodes. Logically speaking, the formalism is simply silent.
In other words, I think there holds a "garbage in, garbage out" principle with regards to probabilities/collapse. You have to introduce something non-trivial in the formalism that gives rise to something akin to a collapse to get definite outcomes with some probabilities when you measure a non-eigenstate. As far as I can see, there is nothing in the argument by Hartle that does this and so it ought to be a foregone conclusion that he cannot derive the Born rule.
Now, if such objections are correct then it still needs to be answered as to what Hartle has actually showed because his result surely is peculiar and interesting!
