Is the Ensemble Interpretation Inconsistent with the PBR Theorem?

[martinbn]

I must say that I am confused. Either I don't understand something (most likely) or what he calls ensemble interpretation is more that an ensemble interpretation, it is ensemble++.

 

[Demystifier]

I must say that I am confused. Either I don't understand something (most likely) or what he calls ensemble interpretation is more that an ensemble interpretation, it is ensemble++.

C++ is C plus object programming. Ensemble++ is ensemble plus object-ive existence.

 

[vanhees71]

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.

So how is then in this example $\lambda$ defined and how do you then conclude that the quantum state is "ontic"? The quantum-mechanical calculation is of course simple and clear, and I still don't see how this example or any other can "disprove" the ensemble interpretation.

 

[vanhees71]

Because the Ballentine interpretation assumes that objective microscopic properties exist, but does not specify what they are. PBR also assume that they exist (they call them $\lambda$) and also don't specify what they are.

No it's very well defined by quantum theory, what's objective: It's the probabilities for the outcome of measurements of any observable, given the state/preparation of the system. "Minimal" means that there's no other objective "reality" than these probabilities described by the quantum state. "Ensemble" means that you can empirically check these probabilities only on an ensemble by statistical evaluation of the measurement outcomes.

Whether or not this is a complete description of nature the minimal interpretation is agnostic about. As for any theory (including classical deterministic theories) it's an empirical question, whether or not it describes all phenomena completely.

The status of QT today is that it does, because there's no experiment in contradiction with the (probabilistic) predictions of QT and nobody has found any "hidden variables" neglected by our present theories. E.g., nobody has ever found an observable which tells us when an individual unstable nucleus decays. All we can predict is some mean lifetime, which can be empirically checked/determined by making a measurement on a large esemble of such nuclei.

 

[Demystifier]

So how is then in this example $\lambda$ defined and how do you then conclude that the quantum state is "ontic"?

It is not defined explicitly, the proof is not constructive. The theorem proves that $\lambda$ with certain property does not exist in the mathematical sense, by assuming that it does and proving a contradiction. It's quite abstract, so it's not so easy to understand it with a typical physicist way of thinking.

 

[Demystifier]

Whether or not this is a complete description of nature the minimal interpretation is agnostic about.

So how do you interpret the Ballentine's claim of incompleteness (post #14)?

 

[martinbn]

... "Minimal" means that there's no other objective "reality" than these probabilities described by the quantum state. ...

What about this

In contrast, the Statistical Interpretation considers a particle to always be at some position in space, each position being realized with relative frequency $|\psi(r)|^2$ an ensemble of similarily prepared experiments.

It is from Ballentine's paper on the statistical interpretation.

 

[Morbert]

It is from Ballentine's paper on the statistical interpretation.

This stood out to me as well. I always thought of the "statistics" in the statistical interpretation as statistics regarding measurement outcomes as opposed to statistics regarding Bell-like beables.

 

[Demystifier]

Maybe for Ballentine "to be" means "to be measured"?
I don't think so, but maybe @vanhees71 could use such an argument.

 

[martinbn]

This stood out to me as well. I always thought of the "statistics" in the statistical interpretation as statistics regarding measurement outcomes as opposed to statistics regarding Bell-like beables.

May be I havn't ready carefully and have taken it out of context. May be in that paragraph he is making the point that it is not like that. May be.

 

[vanhees71]

It is not defined explicitly, the proof is not constructive. The theorem proves that $\lambda$ with certain property does not exist in the mathematical sense, by assuming that it does and proving a contradiction. It's quite abstract, so it's not so easy to understand it with a typical physicist way of thinking.

Ok, I try to understand it again. There must be some meaning in what's contradicted by the quantum formalism to what's described by probability distributions for $\lambda$ which is not defined ;-)). I'm always a bit lost with such presumably "mathematical" proofs with only vaguely defined quantities, which then are supposed to have some philosophical meaning like the contradistinction between ontic and epistemic ;-)). It's strange to have vague definitions in mathematics and proving something about these vague definitions ;-)).

 

[vanhees71]

Maybe for Ballentine "to be" means "to be measured"?
I don't think so, but maybe @vanhees71 could use such an argument.

I don't know, I'm not able to read Ballentine's mind ;-)).

 

[Demystifier]

Ok, I try to understand it again. There must be some meaning in what's contradicted by the quantum formalism to what's described by probability distributions for $\lambda$ which is not defined ;-)). I'm always a bit lost with such presumably "mathematical" proofs with only vaguely defined quantities, which then are supposed to have some philosophical meaning like the contradistinction between ontic and epistemic ;-)). It's strange to have vague definitions in mathematics and proving something about these vague definitions ;-)).

I agree, except that I would not call it vague but abstract. The famous Godel theorems, as well as the Banach-Tarski paradox, are also of this sort.

 

[vanhees71]

What about this

"In contrast, the Statistical Interpretation considers a particle to always be at some position in space, each position being realized with relative frequency an ensemble of similarily prepared experiments."

It is from Ballentine's paper on the statistical interpretation.

This quote I would not sign. In QT an observable has either a determined value (due to preparation) or it has no determined value, because the system is prepared in a state, where the probability for finding some value is non-zero for at least one possible outcome of the measurement.

For me the strength of the statistical interpretation was that it takes Born's rule seriously and states that the only meaning of the quantum state are the probabilities for the outcomes of measurements.

To assume that "a particle always is at some (definite) position in space" would somehow imply that the position vector has always a determined value, no matter in which state the particle is prepared, but this, at least for me, is not what the quantum formalism tells us. It then would immediately imply some HVs which determine this position and thus that the "quantum probabilities" would be only "subjective", i.e., due to incomplete knowledge about the state. Then you'd need an extension of QT to some (according to Bell and the empirical findings about Bell's inequality necessarily non-local) deterministic theory, which however nobody ever has been able to formulate.

 

[Demystifier]

Then you'd need an extension of QT to some (according to Bell and the empirical findings about Bell's inequality necessarily non-local) deterministic theory, which however nobody ever has been able to formulate.

Except Bohm, of course.

 

[Demystifier]

This quote I would not sign.

So you disagree with Ballentine?

 

[vanhees71]

I fear so ;-). I've to read the old RMP paper again. The more one thinks about the foundations the more you change your opinion yourself over the years!

 

[PeterDonis]

Except Bohm, of course.

This quote from the 1970 paper...

In contrast, the Statistical Interpretation considers a particle to always be at some position in space, each position being realized with relative frequency $|\psi(r)|^2$ an ensemble of similarily prepared experiments.

...makes it seem to me like Ballentine himself is a Bohmian!

 

[martinbn]

This quote I would not sign. In QT an observable has either a determined value (due to preparation) or it has no determined value, because the system is prepared in a state, where the probability for finding some value is non-zero for at least one possible outcome of the measurement.

For me the strength of the statistical interpretation was that it takes Born's rule seriously and states that the only meaning of the quantum state are the probabilities for the outcomes of measurements.

To assume that "a particle always is at some (definite) position in space" would somehow imply that the position vector has always a determined value, no matter in which state the particle is prepared, but this, at least for me, is not what the quantum formalism tells us. It then would immediately imply some HVs which determine this position and thus that the "quantum probabilities" would be only "subjective", i.e., due to incomplete knowledge about the state. Then you'd need an extension of QT to some (according to Bell and the empirical findings about Bell's inequality necessarily non-local) deterministic theory, which however nobody ever has been able to formulate.

Can you phrase all that using statistical interpretation language? You talk about the system/particle, its state and the observables as if everything refers to a single object, but the state is the state of the ensemble, not of just one representative of it and so on.

 

[Demystifier]

This quote from the 1970 paper...
...makes it seem to me like Ballentine himself is a Bohmian!

The difference is that Ballentine is agnostic about determinism. Particle can have a position x at each time t, but x(t) can be stochastic (instead of deterministic). An explicit example is the Nelson interpretation.

 

[atyy]

...makes it seem to me like Ballentine himself is a Bohmian!

Yes, both the book and the paper mention Einstein's interpretation, which is why it is plausible to read Ballentine as assuming hidden variables. However, such an assumption ought to be stated clearly, and the variables and their dynamics stated. But Ballentine doesn't do that. And even if one preferred a hidden variables interpretation, it would not justify his criticism of the standard interpretation - since if the hidden variables view were right, it must derive the standard Copenhagen-style interpretation as an effective theory. I dislike his criticism of Messiah, since Messiah discusses the possibility of hidden variables, and says they have not been ruled out, but it appears not possible to test at the moment, and says he will present Copenhagen in the rest of the book - so it's a broad minded view that gives proper weight to Einstein's view.

Incidentally, the paper also has another wrong criticism of the standard interpretation. The paper claims that position and momentum can be simultaneously measured, but in the counterexample he gives, the position and momentum are not canonically conjugate. So like the book there are technically incorrect criticisms of standard physics. And although these might be incidental carelessness, the overall point he is making is a huge point - he is saying that textbook QM is wrong (as opposed to saying that standard textbooks are a little sloppy in their presentation). Incidentally, the error he makes shows he has not understood why the Bohmian and Copenhagen interpretations are consistent - in making the error, he does not use Bohmian trajectories.

 

[atyy]

Ok, I try to understand it again. There must be some meaning in what's contradicted by the quantum formalism to what's described by probability distributions for $\lambda$ which is not defined ;-)). I'm always a bit lost with such presumably "mathematical" proofs with only vaguely defined quantities, which then are supposed to have some philosophical meaning like the contradistinction between ontic and epistemic ;-)). It's strange to have vague definitions in mathematics and proving something about these vague definitions ;-)).

I think one way you can think of it is that the state space of quantum mechanics is not a simplex, However, the state space of classical probability is a simplex. The question is whether it is possible to construct a theory preserving all the predictions of QM (to some accuracy) that has an enlarged state space that is a simplex. [Though I guess this criterion is problematic for continuous variables, since I think the state space is not a simplex for classical continuous variables?]

 

[Demystifier]

I fear so ;-). I've to read the old RMP paper again. The more one thinks about the foundations the more you change your opinion yourself over the years!

Perhaps it's time that you write down a paper on your own interpretation!

 

[Demystifier]

I think one way you can think of it is that the state space of quantum mechanics is not a simplex, However, the state space of classical probability is a simplex. The question is whether it is possible to construct a theory preserving all the predictions of QM (to some accuracy) that has an enlarged state space that is a simplex. [Though I guess this criterion is problematic for continuous variables, since I think the state space is not a simplex for classical continuous variables?]

What do you mean by "simplex"?

 

[atyy]

What do you mean by "simplex"?

A shape with sharp points. Like Fig 1.2 in https://www.researchgate.net/publication/258239605_Geometry_of_Quantum_States.

 

[martinbn]

...The paper claims that position and momentum can be simultaneously measured,...

Can you point to where the claim is, so that we can see the context.

 

[Demystifier]

A shape with sharp points. Like Fig 1.2 in https://www.researchgate.net/publication/258239605_Geometry_of_Quantum_States.

So if I understood it correctly, classical probability space is simplex because the probabilities satisfy $p_i\geq 0$, while the quantum state space is not simplex because the coefficients of superposition do not satisfy $c_i\geq 0$, is that right?

 

[atyy]

So if I understood it correctly, classical probability space is simplex because the probabilities satisfy $p_i\geq 0$, while the quantum state space is not simplex because the coefficients of superposition do not satisfy $c_i\geq 0$, is that right?

I'm not sure off the top of my head, but in corresponds to a classical uncertainty being a unique mix of "pure states" (the complete state that can be assigned to a single system), whereas quantum density matrices don't have a unique decomposition into pure states (ie. preferred basis must be picked out by measurement or decoherence or whatever).

Holevo has some discussion at the start of his book (I'm don't have it with me at the moment).

 

[Morbert]

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.

I don't think all interpretations would model reality with an ontic state space (which as far as I can tell is a classical state space).

PBR: If we partition the $\lambda$ state space into discrete regions, each with a unique label $L_i$, then $L$ is a property of the system since the state $\lambda$ uniquely selects a value $L_i$ of $L$

Consistent Histories: If we perform a projective decomposition of the Hilbert space into orthogonal subspaces, each with a unique label $L_i$, then $L$ is a property of the system, but a state $|\psi\rangle$ will not necessarily select a value $L_i$ unless one of the subspaces of the decomposition is spanned by $|\psi\rangle$, so the state does not uniquely specify all properties.

Typical Copenhagen: If we perform a projective decomposition of the Hilbert space into orthogonal subspaces, each with a unique label $L_i$, then there is a hypothetical measurement of $L$ that can be carried out, which will produce one of the outcomes $L_i$ with a probability uniquely specified by the state.

 

[vanhees71]

Can you phrase all that using statistical interpretation language? You talk about the system/particle, its state and the observables as if everything refers to a single object, but the state is the state of the ensemble, not of just one representative of it and so on.

Maybe, I've my own version of "minimal interpretation". So here I try to very quickly state my point of view:

State: Describes an equivalence class of preparation procedures on a single system. It is represented by the statistical operator $\hat{\rho}$ (positive semidefinite self-adjoint operator of trace 1).

Observable: describes an equivalence of measurement procedures. An observable is represented by a self-adjoint operator. The possible values when measured accurately are the eigenvalues of the operators.

Meaning of states: The meaning and the only meaning of the states is that in a precise measurement the probabilities for the outcomes of an exact measurement of an observable are given by Born's rule or equivalently the all moments of the ditribution are given by $\mathrm{Tr}(\hat{\rho} \hat{A}^n)$ with $n \in \mathbb{N}_0$.

In this way the states are referring on the one hand to single objects (preparation procedure for single objects). On the other hand they don't have much of a meaning for the single object and measurements on a singe object, because the state preparation only determines the probabilities for the outcome of measurements, and these probabilities can be empirically tested only on ensembles of equally prepared systems.

The dynamics of a closed system is described by the usual unitary time evolutions of statistical operators and observable operators (and thus their eigenvectors) after fixing an arbitrary picture of time evolution. The physical meaning is independent of the choice of the picture.

 

[vanhees71]

Perhaps it's time that you write down a paper on your own interpretation!

Adding again one more interpretation? What should this be good for?

 

[Demystifier]

the states are referring on the one hand to single objects ... On the other hand they don't have much of a meaning for the single object

Don't you find it confusing?

 

[Demystifier]

Adding again one more interpretation? What should this be good for?

You would not need to respond to silly questions on this forum, you could just point to your paper. In that way you would have much more time for shut up and calculate.

 

[Fra]

think the ensemble interpretation with non-objective properties would be more-or-less equivalent to QBism

There is no full inside agent theory yet but conceptually the ensemble picture of small subatomic physics seems to conceptually correspond to agents living in the the classical background environment, where they moreover can "communicate" classically and form consensus without "quantum weirdness" and without risk beeing "saturated" by information. Ie. the Agents can make inferences and non-lossy storage. These agents are making inferences and predictions from a "safe" distance, so that we can assume that they themselves are not affected by the backreaction ofthe system they interact with.

So the connecton between ensemble view and agent view makes sense.

/Fredrik

 

[atyy]

Can you point to where the claim is, so that we can see the context.

Section 3.2 and Fig 3 of Ballentine's 1970 article. It is true that the usual uncertainty principle does not refer to simultaneous measurement, but it does not mean that simultaneous measurement of canonically conjugate position and momentum are possible.