Is the wavefunction ontological or epistemological?

Part 5: The PBR Result

A few years ago a Nature paper by Pusey, Barrett, and Rudolf (PBR for short) (http://arxiv.org/abs/1111.3328) was the talk of the town. The paper claims the quantum states cannot be interpreted statistically, but this is an overreach and the proper claim is that the wavefunction cannot be “psi-epistemic”. A very good review on this can be found on Matt Leifer’s blog: http://mattleifer.info/2011/11/20/can-the-quantum-state-be-interpreted-statistically/ and thought provoking posts were written by Lubos Motl: http://motls.blogspot.com/2011/11/nature-hypes-anti-qm-crackpot-paper-by.html and Scott Aaronson: http://www.scottaaronson.com/blog/?p=822

To clarify the intent of the paper, let me quote part of the archive abstract: "we show that any model in which a quantum state represents mere information about an underlying physical state of the system, and in which systems that are prepared independently have independent physical states, must make predictions which contradict those of quantum theory" 

So what does a psi-epistemic wavefunction mean? It means that one can have a probability distribution function over ontic states and it is possible to have overlapping functions over the same ontic states. Using this definition, here is how the proof goes:

Start with two pure states: |0> and |1> and construct a superposition |+> = (|0> + |1>)/sqrt(2)

Then for a tensor product corresponding to the system being prepared in either |0> or the |+> states, consider a cleverly chosen basis of four orthogonal states:

|Blue> = (|0>|1> + |1>|0>)/sqrt(2)

|Red> = (|0>|-> + |1>|+>)/sqrt(2)

|Gold> = (|+>|1> + |->|0>)/sqrt(2)

|Green> = (|+>|-> + |->|+>)/sqrt(2)

where |->=(|0> - |1>)/sqrt(2)

Now we have the following properties:

|Blue> is orthogonal with |0>|0>

|Red> is orthogonal with |0>|+>

|Gold> is orthogonal with |+>|0>

|Green> is orthogonal with |+>|+>

which means that measuring in this  basis would yield either:

the system was not prepared as 00

the system was not prepared as 0+

the system was not prepared as +0

the system was not prepared as ++

Let us paint the picture of the statements above using psi-epistemic definition:

Here we see the red and green overlap of the hypothetical probability distribution over some hypothetical ontical state. The key in in the overlap. This allows to claim an epistemic interpretation.

Let us highlight the probability distribution support for the 4 cases:

Not in |0>|0>:

Not in |0>|+>:

Not in |+>|0>:

Not in |+>|+>:

Now PBR paper claims that the following black area:

generates a contradiction because the 4 vector basis forms a complete orthogonal base and no matter the outcome result, it will generate a contradiction with the assumption of understanding the wavefunction as a probability distribution over some ontic state.

Convinced?

Technically the paper is correct, but can we reason in this fashion in quantum mechanics? The contradiction is based on 2-dimensional Venn diagrams. Potential Danger! 

Suppose the 4 compact support areas are not 2 dimensional objects, and we need to reason in space!

Consider this picture instead:

Here there is no intersection of the 4 areas, and the earlier intersection was an optical illusion stemming from looking at this case from a particular viewpoint (that of classical concepts, and 2-dimensional Venn diagram).

This does not correspond to any quantum system, and it is not a counterexample to PBR, but it is an illustration of the dangers of thinking 2-dimensionally in terms of classical sets.

To improve this 3D picture's agreement with the PBR argument one can add for example depth to the color rectangles and extending the depth until any 2 rectangles begin touching. Also picture this on a torus with the red slab touching and overlapping with both the green and gold slabs, the green slab touching and overlapping with both red and blue, etc. (drawing this goes beyond my ability of using Paint).

There is a key piece of information in favor of PBR however: quantum AND is the same as classical AND. However, in higher dimensions, should we demand that all 4 areas must intersect at the same time? This is one of Lubos' criticism:  measuring “not 00” only eliminates this possibility. Because “unperformed experiments have no results” as Asher Peres put it, this is certainly a valid criticism under appropriate conditions. 

To complete the PBR argument and generate the contradiction you need to regard the ontic states as having definite properties and this in turn allows reasoning with sets and 2D Venn diagrams (as a caveat this does not necessarily mean definite observable values and we make take for example the wavefunction itself as the ontic object). Therefore agreeing with PBR hinges on the very definition of what we might mean by “ontic states”.

In conclusion, does PBR proved that quantum wavefunction is not epistemic? Yes and no: yes, it is not psi-epistemic, but this is not the whole story in terms of epistemic explanations (see Matt's blog post). 

Next time I’ll show that quantum systems cannot have definite observable values before measurement and attempt to get a handle of what we might mean by “objective reality”.

UPDATE

The topic of PBR is subtle and the original post was mildly changed to avoid giving the wrong impression. PBR does not disprove Born's interpretation but the consensus is that it disproves the quantum Bayesian interpretation which contends that the collapse postulate is nothing but a manifestation of information update about a quantum system. If the wavefunction is nothing but subjective degree of knowledge, then before the collapse, the wavefunction has some overlap with the wavefunction after the collapse and is subject to PBR's assumptions. Hence PBR rejects the quantum Bayesian interpretation.

Both Chris Fuchs and myself disagree with this analysis (for different reasons), and I contend that the Bayesian interpretation is "successfully reasoning consistently about an inconsistent system" because the collapse postulate is inconsistent with quantum mechanics. As such the wavefunction in this interpretation does not satisfy the "ontic state requirement" of having definite properties since technically the wavefunction in this interpretation does not rigurously exist.