Monday, August 12, 2013

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) ( 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: and thought provoking posts were written by Lubos Motl: and Scott Aaronson:

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.


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”.


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. 


  1. Very interesting post. I wonder if, without regard to the PBR theorem, experiments can state that the quantum wavefunction can be considered as ontological. Please, take a look to the following links to get an answer:–May-the-Bohr-Model-rest-in-peace


  2. Thank you for the links, they were very informative. I think in part due to PBR, the majority of physicists working in quantum foundations are "psi-ontologists" (with the noted exception of Chris' group). I don't really know what "psi-ontology" is but I am advocating for a mild ontological interpretation called "elliptic ontology" (please see all the posts in this series).

    I don't think the ontological interpretation can be ever settled by experiments because an interpretation is a meta-mathematical concept. What can settle this debate are the efforts to recover quantum mechanics from first physics principle and after the project is complete creating a paradigm out of it. (the project is expected to be finalized in a few years- see for the first piece, I have more unpublished results and I am working on the remaining problems).

    What I am certain at this time is that the interpretation cannot be epistemic (and I have an unpublished result which is much stronger than PBR) but the key question is what kind of ontological interpretation is appropriate for the wavefunction.

    The typical mistake (which is made to my big surprise even by well known experts in quantum foundations - I won't name names) on wavefunction ontology is to forget that the wavefunction doe not exist in space-time, but in configuration space. What does this mean? For a 1-particle wavefunction the configuration space *is* the spacetime but for N-particle, it is not. For example a 2 particle wavefunction is written as: psi(x1, x2, t) which is not of the form function (x,t).

    One good example of this configuration space business is Mott's problem ( The wavefunction is spherically symmetric, but the particle tracks are straight lines. There is no contradiction once one realize that the wavefunction is spherically symmetric in configuration space not in real space and Mott showed mathematically how this demands the the tracks to be straight lines.

  3. Thank you very much for your quick and mindful reply. Here other publications in favour of the ontological interpretation:

    Also, I found a work which defined a deterministic quantum protocol:

    Nevertheless, other recent publications enforced a psi-epistemic interpretation:

    In your opinion, is ontic vs epistemic debate only a phylosophical conjecture or is there any real possibility to rule out one of the two interpretations?

  4. Thank you for the references above, I will study them in detail in the following days.

    On the epistemic vs. ontologic, I do have an argument (unpublished and I won't elaborate on it) which rules out the epistemic interpretation in a rather definite way.

    Until I do write a paper on it, I can provide a handwaving physical argument: consider a Rindler horizon ( and a black hole event horizon. Once the horizon is crossed, there is no communication possible. If QM is only about information, there should be no physical difference between the two horizons. But physically there is a difference.

    I won't attempt to solve the black hole information problem (and I don't know how anyway), but here is the problem with black holes: QM enforces no clone, and the trapped information must escape in terms of Hawking radiation. Escaping information come at odds with no cloning. Now for the Rindler horizon there is no information duplication or destruction problem, and stopping the acceleration gets us back to sanity. Again if QM is only about information and nature is QM at core (it from bit), the physics at the horizons should be identical but it is not.

    The argument is only handwaving because first you must have a full theory of quantum gravity. Then one has to show the physical difference is not due to gravity.

    I don't know how to do that, but I know how to transform this argument in another argument where gravity is not present. The math is highly nontrivial.

  5. Congratulation for your wonderful work. It sounds very impressive. Thanks for the contribution given to science. Best wishes for your scientific achievements. I'll often take a look to your website for promising updates.

    Best Regards

  6. Thank you for your kind words. I wish my progress would be faster, but I am breaking new mathematical ground (see for example my Grothendieck group work) and it takes time to fully work out and understand the implications.

    Please stay tuned to my blog, I am following Baez's model of "this week's finds" but I am doing it in series instead. I'll try to keep up uploading new posts of substance every week.

  7. "If QM is only about information, there should be no physical difference between the two horizons. But physically there is a difference."

    Perhaps because a Rinder Horizon is a thought experiment and an Event Horizon is not?

    1. Rindler horizon is quite real and it means you cannot obtain information beyond it. An accelerated observer experiences the Unruh radiation which some claims was already observed:

      but this is disputed.