PeterDonis said:
His whole book is about figuring out objective rules for the robot to follow for a given problem. He never talks about different robots using different rules for the same problem; he clearly believes that for any given problem, there is one correct set of rules, and that's the set he's looking for.
Not 'there is' but 'there should be'! Jaynes argues about rules robots
should follow, rather than the rules they
actually follow. Jaynes' rules are normative (desiderata), not descriptive (facts). Moreover, even the rules he gives all depend on the prior probability assignment, which is subjective, according to Jaynes' own testimony on p.44:
Jaynes (my italics) said:
In the theory we are developing, any probability assignment is necessarily ‘subjective’ in the sense that it describes only a state of knowledge, and not anything that could be measured in a physical experiment. Inevitably, someone will demand to know: ‘Whose state of knowledge?’ The answer is always: ‘That of the robot – or of anyone else who is given the same information and reasons according to the desiderata used in our derivations in this chapter.’
Anyone who has the same information, but comes to a different conclusion than our robot, is necessarily violating one of those desiderata.
But reality is so complex that many things require qualitative judgment - something that cannot be formalized since (unlike probability, where there is a fair consensus about the basic rules to apply) there is no agreement among humans about how to judge. This is why different scientists confronted with the same data can come to quite different conclusions. Violating these desiderata is a necessity. I want to have a philosophy of probability (and of quantum physics) that reflects actual practice, not a wish list.
PeterDonis said:
I don't see how two scientists that are both using the exact same Hilbert space for a given quantum tomography experiment could differ in their computation of P(H|X) for any H.
They differ in the results whenever they differ in the prior. The prior is by definition a probability distribution hence subjective = robot-specific (in Jaynes' scenarios), a state of the mind of the robot. No two robots will have the same state of the mind unless they are clones of each other, in every detail that might affect the course of their computations.
It is a truism that the states of the mind of two scientists is far from being the same. Scientists are individuals, not clones.
PeterDonis said:
when you talk about the estimates of density matrix parameters from the data being objective in the sense of all scientists involved agreeing on them. That agreement will only happen if they all follow the same rules in doing their computations.
Agreement only means agreement to some statistical accuracy appropriate for the experiments analyzed. I didn't claim perfect agreement.
In my paper I talk about the
standard statistical procedures (non-Bayesian, hence violating the desiderata of Jaynes): Using simple relative frequencies to approximate the probabilities that enter the quantum tomography process, and then solving the resulting set of linear equations. In the case of N independent measurements) the tomography results will have (by the law of large numbers) an accuracy of ##O(N^{-1/2})## with a factor in the Landau symbol depending on the details of the statistical estimation procedure and the rounding errors made. A lot of ingenuity goes into making the factor small enough so that reasonably accurate results are possible for more than the tiniest system, which explains why scientists using the same data but different software will get slightly different results. But the details do not matter to conclude that in principle, i.e., allowing for arbitrarily many experimental and exact computation, the true state (density operator) can be found with as close to certainty as one likes. This makes the density operator an objective property of the stationary quantum beam studied, in spite of the different results that one gets in actual computations. The differences are comparable in nature of the differences one gets when different scientists repeat a precisely defined experiment - measurement results are well-known to be not exact, but what is measured is nevertheless thought of (in the model) as something objective.
PeterDonis said:
If a physicist were to tell you he doesn't agree with your density matrix parameter estimates from quantum tomography data, you would expect him to give some cogent physics reason like he thinks you're using the wrong Hilbert space for the system.
Yes, and the cogent reason is that he uses different software and/or weighted the data differently because of this or that judgment, but gets a result consistent with the accuracies to be expected. There are many examples of scientists measuring tabulated physical constants or properties, and the rule is that different studies arrive at different conclusions. Even when analyzing the same data.
No competent physicist would use a
wrong Hilbert space, but there are reasons why someone may choose a different Hilbert space than I did in your hypothetical setting: For efficient quantum tomography you need to truncate an infinite-dimensional Hilbert space by a subspace of very low dimensions, and picking this subspace is a matter of judgment and can be done in multiple defensible ways. Results differ. With time, some methods (and details inside the methods) prove to give more accurate or more robust results, and these become standard until superseded by even better methods.
Quantum chemical calculations of ground state energies of molecules are a well-known example where depending on the accuracy wanted you need to choose different schemes, and
results are never exactly reproducible unless you use the same software with the same parameters, and in case of quantum Monte Carlo calculations also the same random number generator and the same seed.
PeterDonis said:
explicit purpose to remove "arbitrariness" in assigning prior probabilities.
Jaynes does not succeed in this. There is a notion of noninformative prior for certain classes of estimation problems, but
this gives a good prior only if (from the point of view of a frequentist) it resembles the true probability distribution. (Just as in quantum tomography, 'true' makes sense in cases where one can in principle draw arbitrarily large samples of independent realizations.) The reason is that no probability distribution is truly noninformative, so if you don't have past data (or extrapolate from similar experiences) whatever prior you pick is pure prejudice or hope.