Is the Ensemble Interpretation Inconsistent with the PBR Theorem?

[Demystifier]

[Moderator's note: Spun off from another thread due to topic and subforum change.]

I think Ballentine's interpretation is ruled out by the PBR theorem. Maybe we could discuss that?

 

[vanhees71]

Ballentine does not assume that the probabilities in QT are due to incomplete knowledge (in the spirit of hidden-variable theories), which is what the PBR theorem disproves, right? So why is Ballentine's interpretation fuled out by the PBR theorem?

 

[Demystifier]

Ballentine does not assume that the probabilities in QT are due to incomplete knowledge (in the spirit of hidden-variable theories), which is what the PBR theorem disproves, right? So why is Ballentine's interpretation fuled out by the PBR theorem?

No, that's not what PBR theorem disproves. Ballentine assumes that
1) The quantum state does not describe properties of a single object, but only probabilities in an ensemble.
2) The properties of a single object are objective, i.e. do not depend on someone's knowledge about them.
The PBR theorem shows that if 2) is right then 1) isn't; the state describes not only the probabilities in an ensemble, but also tells something about properties of a single object.

 

[vanhees71]

Ok, then I have to learn about the meaning of the PBR theorem better. I've great difficulties with the language concerning philosohpical terms like epistemic and ontic without a mathematical definition of what's meant by this terms. Is there a formulation of the PBR theorem and its proof in terms of clear mathematical statements? In the Nature Physics paper, I don't understand in which state the system is assumed to be prepared in before making the projective Bell measurement projecting to the states mentioned in (1). If I read the text immediately before the equation, in my understanding all I can say before the measurement is that it is in the mixed state (using the maximum-entropy method to determine it from the incomplete information given)
$$\hat{\rho}=\frac{1}{4} (|0,0 \rangle \langle 0,0| + |0,+ \rangle \langle 0,+| + |+,0 \rangle \langle +,0| + |+,+ \rangle \langle |+,_+\rangle.$$
Also what means "overlap"? The paper is too vague for me to understand its content properly.

I'd formulate the first statement within the minimal interpretation somewhat differently:

(1) The meaning of the quantum theoretical state (given by an equivalence class of preparation procedures) is given by Born's rule, i.e., there is only probabilistic information about the outcome of measurements on the so prepared system. This can be experimentally checked only on an ensemble of equally prepared systems.

Concerning 2): Does the PBR theorem imply that the properties of a single object are not objective, i.e., depend on someone's knowledge about them?

That doesn't make much sense to me. According to the minimal standard interpretation, the single object in an ensemble of the kind described in (1) is prepared in some (pure or mixed) quantum state, which has nothing to do with the knowledge an individual has about it. It's in this state due to the preparation procedure and not due to any knowledge the one or other person has about its state. The state is an objective property of the single object, defining an ensemble upon which observables can be measured and the probabilistic predictions of QT for this state can be checked by statistics.

 

[Demystifier]

Ok, then I have to learn about the meaning of the PBR theorem better. ...

Maybe my talk can be of some help:
http://thphys.irb.hr/wiki/main/images/c/cb/PBR.pdf

 

[Demystifier]

According to the minimal standard interpretation, ... The state is an objective property of the single object,

Maybe that is so according to the "standard" interpretation, but that is not so according to the Ballentine interpretation.

 

[Demystifier]

I've great difficulties with the language concerning philosohpical terms like epistemic and ontic without a mathematical definition of what's meant by this terms. Is there a formulation of the PBR theorem and its proof in terms of clear mathematical statements?

In the PBR paper, the meaning of "ontic" and "epistemic" are defined precisely and mathematically.

 

[martinbn]

No, that's not what PBR theorem disproves. Ballentine assumes that
1) The quantum state does not describe properties of a single object, but only probabilities in an ensemble.
2) The properties of a single object are objective, i.e. do not depend on someone's knowledge about them.
The PBR theorem shows that if 2) is right then 1) isn't; the state describes not only the probabilities in an ensemble, but also tells something about properties of a single object.

What do you mean by two? What is a property here?

 

[PeterDonis]

Ballentine assumes that
...
2) The properties of a single object are objective,

Where does Ballentine assume this? No such assumption is required for the ensemble interpretation. That interpretation requires that the probabilities of various measurement results be objective (i.e., different experimenters setting up the same ensemble will get the same results), but that says nothing about any properties of a single object.

 

[Demystifier]

Where does Ballentine assume this? No such assumption is required for the ensemble interpretation.

Maybe Ballentine never says explicitly that properties of a single object are objective, but it seems to me that it is implicit in the Ballentine's way of thinking. I think the ensemble interpretation with non-objective properties would be more-or-less equivalent to QBism, in which case, I think, Ballentine wouldn't conclude that Bell theorem proves nonlocality (which he does).

Anyway, if we accept that properties of a single object may or may not be objective in the ensemble interpretation, then we can talk about two versions of the ensemble interpretation and argue that PBR rules out the first version. Either both the state and the single object properties are objective (Bohm, GRW, many worlds), or neither is objective (QBism).

 

[Demystifier]

What do you mean by two? What is a property here?

E.g. the detector clicked once, even if there was nobody to hear that.

 

[Morbert]

A physicist carriers out some preparation procedure, and describes the preparation with the trace-class operator $|\psi\rangle\langle\psi|$.

i) PBR authors say this preparation procedure will result in the physical system being in a state $\lambda$ with probability $\mu_\psi(\lambda)$
ii) PBR authors say a property of a system is associated with a partition of a classical state space into disjoint regions

My question: Why would Ensemble proponents accept i) and ii)? They might dismiss i) out of hand, and they might instead associate a property with a partition of a Hilbert space into subspaces with orthogonal projectors, eliminating the need for the classical state space in ii)

 

[martinbn]

E.g. the detector clicked once, even if there was nobody to hear that.

How is this a property of the object!

This whole thing is soo vague that it makes it impossible to talk about. Everyone will talk about something differerent, whatever he understands by these terms.

 

[Demystifier]

Here I present further evidence that Ballentine assumes that the individual properties are objective. For that purpose I first quote some parts of his famous 1970 paper in Reviews of Modern Physics. The boldings are mine.

"(I) The Statistical Interpretation, according to which a pure state (and hence also a general state) provides a description of certain statistical properties of an ensemble of similarily prepared systems, but need not provide a complete description of an individual system."

"(II) Interpretations which assert that a pure state provides a complete and exhaustive description of an individual system (e.g., an electron)."

"It is a major aim of this paper to point out that the hypothesis II is unnecessary for quantum theory, and moreover that it leads to serious difficulties."

"Thus a momentum eigenstate (plane wave in configuration space) represents the ensemble whose members are single electrons each having the same momentum, but distributed uniformly over all positions.!"

"Physical systems which have been subjected to the same state preparation will be similar in some of their properties, but not in all of them (similar in momentum but not position in the first example)."

"Thus it is most natural to assert that a quantum state represents an ensemble of similarily prepared systems, but does not provide a complete description of an individual system."

Now let me comment those quotes. Ballentine explicitly claims that the state is not a complete description on an individual system. Implicitly, I think he is saying that there is some additional stuff. In the case of a momentum eigenstate that he mentions, this additional stuff would be the particle position. He does not say whether this additional stuff is objective or subjective, but it seems pretty clear to me that he means objective. If it was subjective then it would depend on the subject, so I think he would not associate it with a single electron.

Did I prove that Ballentine assumes that individual properties are objective? I did not. But I think I provided a compelling evidence for that.

 

[Demystifier]

How is this a property of the object!

"Detector clicked" is a property of the detector.

 

[martinbn]

"Detector clicked" is a property of the detector.

Yes, but not of the quantum system, say the particle that you want to measure.

 

[Demystifier]

A physicist carriers out some preparation procedure, and describes the preparation with the trace-class operator $|\psi\rangle\langle\psi|$.

i) PBR authors say this preparation procedure will result in the physical system being in a state $\lambda$ with probability $\mu_\psi(\lambda)$
ii) PBR authors say a property of a system is associated with a partition of a classical state space into disjoint regions

My question: Why would Ensemble proponents accept i) and ii)? They might dismiss i) out of hand, and they might instead associate a property with a partition of a Hilbert space into subspaces with orthogonal projectors, eliminating the need for the classical state space in ii)

First, I don't see how an ensemble proponent could dismiss i). Just the opposite, I think i) is one of the main assumptions of the Ballentine's interpretation.

Second, I don't see that ii) is essential for the PBR theorem. If I'm wrong, can you pinpoint to the exact place where it is said?

 

[Demystifier]

Yes, but not of the quantum system, say the particle that you want to measure.

Did I say the opposite?

 

[martinbn]

Did I say the opposite?

Then my question remains. Why did you give me this example about the detector? Give me an example about the quantum system.

 

[vanhees71]

Maybe my talk can be of some help:
http://thphys.irb.hr/wiki/main/images/c/cb/PBR.pdf

I'll have a look at it. Does it explain which state is prepared in the example in the paper just before Eq. (1) and what $\lambda$ and "overlap" is supposed to mean for this example?

 

[Demystifier]

I'll have a look at it. Does it explain which state is prepared in the example in the paper just before Eq. (1) and what $\lambda$ and "overlap" is supposed to mean for this example?

It does, but perhaps not in a way you would like. You will see, so if you will have additional questions you can ask.

 

[Morbert]

First, I don't see how an ensemble proponent could dismiss i). Just the opposite, I think i) is one of the main assumptions of the Ballentine's interpretation.

Second, I don't see that ii) is essential for the PBR theorem. If I'm wrong, can you pinpoint to the exact place where it is said?

Do you have a link to the Ballentine literature? I'll take a look.

Re/ PBR: See e.g. figure 1 caption here: https://www.nature.com/articles/nphys2309 They define a property by considering a collection of distributions and note that these mark out properties if their supports are disjoint. Maybe all these supports don't span the entire state space (which would be odd to me), but otherwise this seems to mark out a partition of the state space into disjoint regions.

[edit]- Also see this part in the text: " As the energy is a physical property of the system, different values of the energy E and E 0 correspond to disjoint regions of phase space "

 

[Demystifier]

Then my question remains. Why did you give me this example about the detector? Give me an example about the quantum system.

That would depend on the interpretation. In the Bohmian one, it would be particle positions.

 

[vanhees71]

Here I present further evidence that Ballentine assumes that the individual properties are objective. For that purpose I first quote some parts of his famous 1970 paper in Reviews of Modern Physics. The boldings are mine.

"(I) The Statistical Interpretation, according to which a pure state (and hence also a general state) provides a description of certain statistical properties of an ensemble of similarily prepared systems, but need not provide a complete description of an individual system."

"(II) Interpretations which assert that a pure state provides a complete and exhaustive description of an individual system (e.g., an electron)."

"It is a major aim of this paper to point out that the hypothesis II is unnecessary for quantum theory, and moreover that it leads to serious difficulties."

"Thus a momentum eigenstate (plane wave in configuration space) represents the ensemble whose members are single electrons each having the same momentum, but distributed uniformly over all positions.!"

"Physical systems which have been subjected to the same state preparation will be similar in some of their properties, but not in all of them (similar in momentum but not position in the first example)."

"Thus it is most natural to assert that a quantum state represents an ensemble of similarily prepared systems, but does not provide a complete description of an individual system."

Now let me comment those quotes. Ballentine explicitly claims that the state is not a complete description on an individual system. Implicitly, I think he is saying that there is some additional stuff. In the case of a momentum eigenstate that he mentions, this additional stuff would be the particle position. He does not say whether this additional stuff is objective or subjective, but it seems pretty clear to me that he means objective. If it was subjective then it would depend on the subject, so I think he would not associate it with a single electron.

Did I prove that Ballentine assumes that individual properties are objective? I did not. But I think I provided a compelling evidence for that.

I think indeed the trouble is the word "complete" in (I). I always understood the ensemble interpretation such that

(a) The state of the system is a description of the preparation procedure. The preparation is always for a single system and thus the state is objectively associated with the single system. A complete preparation can be achieved by a projective measurement of a complete set of compatible observables. Then the system is in the pure state $\hat{\rho}=\hat{P}(o_1,\ldots,o_j)=|o_1,\ldots,o_n \rangle \langle o_1,\ldots,o_n|$ when $O_j$ provide the complete set of compatible observables which are represented by the corresponding mutually commuting self-adjoint operators $\hat{O}_j$ and the uniquely determined projectors $\hat{P}(o_1,\ldots,o_j)$ with $|o_1,\ldots,o_n \rangle$ being a common eigenbasis of the operators $\hat{O}_j$.

(b) The only meaning of the state (a statistical operator) are the probabilities for the outcome of measurements given by the Born rule and as such this meaning refers to an ensemble of equally prepared single systems.

Whether or not this description is complete, is neither implied nor disproved but subject to empirical tests of the theory. Today, there's no hint of incompleteness, i.e., nobody has ever found examples, where the randomness of the outcome of measurements is due to missing information about hidden variables.

 

[martinbn]

That would depend on the interpretation. In the Bohmian one, it would be particle positions.

For the ensemble interpretation, the one the thread is about.

 

[vanhees71]

It does, but perhaps not in a way you would like. You will see, so if you will have additional questions you can ask.

I glanced over it: What I don't understand what's the meaning of $\lambda$ in the example being discussed starting from slide 16 (the same problem I have in understanding the quoted PBR paper). One has to specify the prepared state and the meaning of $\lambda$. Only then the $\mu(\lambda)$ and the overlap is defined within QT, or is there some other assumption like in Bell's theorem, where you assume a local deterministic HV theory (with a well-defined meaning!)?

 

[Demystifier]

Do you have a link to the Ballentine literature? I'll take a look.

https://www.informationphilosopher.com/solutions/scientists/ballentine/PR70.pdf

Re/ PBR: See e.g. figure 1 caption here: https://www.nature.com/articles/nphys2309 They define a property by considering a collection of distributions and note that these mark out properties if their supports are disjoint. Maybe all these supports don't span the entire state space (which would be odd to me), but otherwise this seems to mark out a partition of the state space into disjoint regions.

[edit]- Also see this part in the text: " As the energy is a physical property of the system, different values of the energy E and E 0 correspond to disjoint regions of phase space "

I think it would be very strange to deny that. Perhaps consistent-histories interpretation denies that (I'm not sure about that), but other interpretations don't.

 

[Demystifier]

For the ensemble interpretation, the one the thread is about.

The trouble with the ensemble interpretation is that it is not explicit about it.

 

[Demystifier]

Today, there's no hint of incompleteness, i.e., nobody has ever found examples, where the randomness of the outcome of measurements is due to missing information about hidden variables.

Ballentine talks about incompleteness in a different sense.

 

[Demystifier]

or is there some other assumption like in Bell's theorem, where you assume a local deterministic HV theory (with a well-defined meaning!)?

Yes, $\lambda$ is something like that in the Bell theorem. But it is not assumed that it is deterministic and local. It is only assumed that it is ontic, in the sense defined mathematically (but quite abstractly) in the paper.