Understanding the Pusey-Barrett-Rudolph (PBR) Paper and Possible Experiments to Test It

Question

I was kindly pointed to the PBR paper by one of the other members here recently and have been trying to digest it. As one might imagine I have a wide range of questions but I'll try to keep things concise and succinctly answerable as best I can.

1. First, starting with the basics, what I read the paper as claiming to prove is that if two systems have the same complete "real" or "physical" state λ then they must have the same quantum state. It then follows through basic logic that if they have different quantum states they must have different physical states. Finally - and I'm projecting here - this means that since measurement / wavefunction collapse alters the quantum state, it must also alter the physical state. Hence, wavefunction collapse is ontic, representing a real physical change in the physical state of the system, and not merely epistemic, representing new information being added to the probabilistic model. However, it does NOT prove that systems that share the same quantum state must share the same physical state, - in other words it doesn't actually say anything about the existence or not of hidden variables. Do I have all this right so far?

2. Again just to make sure I'm understanding it right, I'll list out the actual thought experiment as I follow it. First they assume that it's possible in principle for two physically identical (in all relevant respects) systems to be produced which have different epistemic quantum states (which would contradict the notion of the quantum state being ontic for reasons stated above). Then they propose two devices, each of which can be set to produce particles in either the |0> or |+> quantum states, with |+> being a superposition of |0> and |1>. They further propose that a detector somehow measures the quantum states of the incoming particles. If the generating devices produce particles in the "wrong" quantum state, even infrequently, then the detector will likewise detect the "wrong" quantum states. Again do I have this right so far?

3. Assuming I'm right so far, then where they lose me is how they measure the quantum state as opposed to a real property like spin. I'd guessed this would be by measuring real properties of a large sample of particles coming out of the machines after they'd both been set. So for example if we say |0> means spin-x-down and |+> means spin-x-down|spin-x-up, then if we set the generator to |0> we expect 100% of the electrons to be detected as spin-x-down and if we set it to |+> we expect it to be 50% x-up and 50% x-down. However, it's not clear to me what their detector is actually measuring, as Fig. 2 labels the four possible outcomes as "Not 00", "Not 0+", etc. rather than "00", "0+", etc. It also states that "the measurement is an entangled measurement ... project[ed] onto ... four orthogonal states" which are the respective inverses of the four actual possible states. Again this is where they lose me as I don't understand how entanglement fits into this given that they expressly state the preparations are independent. I also don't follow how a real detector would choose one of the four outcomes. So, in a real PBR experiment what exactly would be measured and how is one of the four outcomes ("Not 00", etc.) actually arrived at? Pointing me to a paper that proposes a real-world experiment to test PBR would be super helpful here if there is one.

4. Lastly, the argument involves a combination of definite states (|0> and |1>) and superposition states (|+> and |->). However, I think someone arguing that the quantum state is epistemic would take it as a given that definite states (indeed ONLY definite states) have physical meaning. What we are interested in is whether superposition states do, and specifically whether the change of quantum state from indefinite to definite (aka collapse) actually changes the physical properties of the entire system or merely elucidates the underlying definite state that was there all along but we were just ignorant of. In the latter interpretation, the physical state would indeed be compatible with an infinite number of quantum states - one definite based on the objective reality, all others indefinite due to incomplete knowledge. Furthermore, I don't think even the most ardent believer in quantum epistomology would question that even if a physical state could be compatible with multiple quantum states, it could never be compatible with multiple orthogonal states. Is mixing definite and indefinite states here - and more specifically the reliance on zero-probability outcomes which by definition require definite states or orthogonal indefinite states - not a weakness in the argument and also the ability to test this empirically?

So, to summarize, if we were to change the key assumption to - two systems with the same physical state λ can support multiple non-orthogonal indefinite quantum states though they must share the same definite quantum states - would PBR still refute this assertion? Or to put it more simply, must all quantum states be considered either ontic or epistemic? Cannot some quantum states be more "real" than others?

Answer 1

1. Yes, that seems accurate to me. There are essentially three (not all mutually exclusive) possibilities of interest here. (1) If one physical state corresponds to several quantum states, then the quantum state encodes our ignorance about what the true physical state actually is. (2) If multiple physical states correspond to the same quantum state, then there must be hidden variables at play beyond the quantum state which determine the physical state. (3) If each physical state corresponds to one and only one quantum state, then there are no hidden variables and the quantum state encodes the full physical state in a unique way. The PBR paper addresses and rules out the first possibility (modulo assumptions). As you say, it does not imply that each quantum state corresponds to a unique physical state, and therefore is agnostic regarding the existence of hidden variables.
2. The four possible quantum states which are sent to the detector are $$|0\rangle\otimes|0\rangle \qquad |0\rangle\otimes|+\rangle \qquad |+\rangle\otimes|0\rangle \qquad |+\rangle\otimes|+\rangle$$ Assume that $|0\rangle$ corresponds to a physical state $\lambda$ with probability $q_0$, and that $|+\rangle$ corresponds to $\lambda$ with probability $q_1$. Let $q=\mathrm{min}(q_0,q_1)$. Then there exists a physical state $\Lambda$ (defined by each particle being in the physical state $\lambda$) which is produced by each of the four quantum states listed above with probability at least $q^2>0$.
3. The problem with this is that we can design a measurement device whose outcomes are $$\matrix{|\xi_1\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle\otimes|1\rangle + |1\rangle\otimes|0\rangle\right) \\ |\xi_2\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle\otimes|-\rangle + |1\rangle\otimes|+\rangle\right) \\ |\xi_3\rangle = \frac{1}{\sqrt{2}}\left(|+\rangle\otimes|1\rangle + |-\rangle\otimes|0\rangle\right) \\ |\xi_4\rangle = \frac{1}{\sqrt{2}}\left(|+\rangle\otimes|-\rangle + |-\rangle\otimes|+\rangle\right)}$$ But the outcome $\xi_1$ is impossible (as per the predictions of QM) if we send in the state $|0\rangle\otimes |0\rangle$. This is why their $\xi_1$ is labeled as "Not 00" in their diagram. But since that quantum state corresponds to the physical state $\Lambda$ with probability at least $q^2$, that means that the outcome of $\xi_1$ must be impossible when the system is in the state $\Lambda$ (otherwise, $\xi_1$ would occur at least occasionally). Similar reasoning shows that $\xi_2,\xi_3,$ and $\xi_4$ must also all be impossible in the state $\Lambda$. But this exhausts all of the possible measurement outcomes of the detector, and yields a contradiction (i.e. with probability $q^2$, none of the possible measurement outcomes can occur).
4. There is no such thing as definite states vs. superposition states. $|0\rangle$ and $|1\rangle$ are states of definite spin along $\hat z$, but superpositions if we measure spin along $\hat x$. Given any quantum state, there are an infinity of observables for which that state is a definite eigenstate, and an infinity of observables for which that state is a superposition of definite eigenstates. It is therefore not meaningful to categorize states as "definite" or "indefinite," nor to predicate the epistemic nature of the wavefunction on one of those two labels.

Experimental tests of PBR can be found here and here - searching for references and citations will lead you to more. Both experiments require the more sophisticated treatment presented towards the end of the original paper (and expanded upon in later papers) which allows for experimental noise and imperfect measurements.

And of course, the PBR theorem rests on assumptions and is therefore subject to loopholes which circumvent its conclusion. It's up to you whether you think those assumptions are reasonable; searching for more information about them may lead you to further theoretical or experimental results which help to close the corresponding loopholes.