Quantum Bayesianism and contradictory preditions of two agents

Question

In quantum Bayesianism (QBsim) interpretation, the wave function $| \psi \rangle$, or density operator $\hat{\rho} = | \psi \rangle \langle \psi |$, is not objective. It is instead interpreted as the coding tool of statistical knowledge of each individual agent (or: observer). This is so far my understanding of the QBsim. In addition I also assume that, the QBism wouldn't provide new rules for predicting quantum phenomena. Assuming the above understanding is accurate and I haven't missed important points of the QBsim, I have a question about it.

Using the famous example in Feynman's book, for the double-slit experiment of an electron, suppose that there are simultaneously two independent agents:

1) the first agent sets up the measurement device near two slits, detecting which path the electron goes by (assuming the device does not destroy the electron, instead disturbs the electron inevitably).

2) the second agent sets up an array of "clicking" device on the screen of far-field, which records the electron's interference pattern.

Then the two agents independently predict whether there is an interference pattern on the screen or not. Let the two agents be independent and not communicate about the wave-function information. How can predictions of the two agents be consistent with the subjective nature of wave functions? Or put it in another way:

1) if the first agent predicts correctly, which appears to be obvious, and

2) the second agent predicts wrongly, which appears to be due to the uninformed "state" knowledge of this agent,

would the contradiction recover the "objectivity" of two agent's wave functions? More specifically, the first agent's "state" knowledge "collapsed" to the correct (or: objective) wave function, while the second agent was ignorant and possessed the wrong wave function. I believe I could write all above reasoning in the usual quantum formalism. So I am wondering if I misunderstood the QBism, or otherwise this example reveals a fundamental problem in the QBism interpretation.

EDIT

The motivation for my question came from my reading Englert's paper:

http://www.physics.nus.edu.sg/~phyebg/arXiv.1308.5290v2.pdf

where on page 8 (section 6) it is talking about the state reduction, which is interpreted as the observer's knowledge update. I just realized that, my original question is not appropriate concerning of what to predict for the two agents. But the point should still be valid:

1) from the first agent experience, the state reduced to a mixture of two paths:

$(|path_1, A_1\rangle \langle path_1, A_1| + |path_2, A_2\rangle \langle path_2, A_2|) / 2$ (1)

2) while from the second agent experience, the state evolved to an entanglement of two paths (before the electron hits the screen).

$(|path_1, A_1\rangle + |path_2, A_2\rangle) / \sqrt{2}$ (2)

where $A_i$ (i = 1, 2) refers to the state of observer 1's detector.

those two states are two different mathematical objects, which are considered "subjective" for each agent. Now it is true that both agents make the same prediction about "no-interference" on the screen. However, would it possible for the two agents to make contradictory predictions about other experiments - since the states in eq. (1) and (2) are two different mathematical objects?

Answer 1

QBism is much more than that: the second observer's subjective experience contains the two slits, the measuring device installed by the first observer, the first observer himself and the rest of the world. Only by taking into account all this data can he estimate probabilities properly.

But even if their experiences disagree, there is no way they could know that. Because each one thinks of himself as supreme and the other one being just quantum matter.

I would like to quote David Mermin here:

I would say that each user of quantum mechanics builds a personal representation of reality based entirely on his experience. Since what you tell me about your experience is part of my experience, we can expect considerable overlap in our different models of reality.

Note that "considerable overlap" does not mean that everything we feel should be the same.

Answer 2

This is an answer to the second version of the question, namely

would it possible for the two agents to make contradictory predictions about other experiments

The answer is no, assuming that your original distinction between the two observers holds. To be precise, this distinction is that observer 1 has access to the state of the detector after the measurement, while observer 2 does not. This means that 2 can trace over the states of the detector, yielding the state $$\rho_2 = \mathrm{Tr}_A\; |\psi_2\rangle\langle \psi_2| = \frac{1}{2}(|path_1\rangle\langle path_1| + |path_2\rangle\langle path_2|).$$

It is easy to verify that observer 1 obtains the same state when tracing over the detector. Therefore, both observers agree on all measurements that could be performed on the $path$ variables only, including whether or not an interference pattern can be found on the screen. In other words, no measurement on the $path$ variables alone can distinguish the two states.

It is only possible to distinguish the two states if you measure some global property of the particle+detector system, i.e. look at correlations between path measurements and detector measurements. But by assumption, observer 2 does not have access to the results of observer 1's measurements.

Answer 3

You might like https://arxiv.org/abs/hep-th/0110253 and the references here, especially the discussion of the simple interferometer on page 11...

http://physics.bu.edu/~youssef/quantum/quantum_refs.html