Assumptions of the Bell theorem

[PeterDonis]

For state reduction

Which is dynamics, not kinematics.

In classical probability this map gives the Bayesian update rule

Which is dynamics, not kinematics.

If you are going to exclude unitary evolution from kinematics (which, as I said, I can see would make sense), then you have to also exclude any other kind of "evolution" of the state from kinematics.

 

[PeterDonis]

In quantum information theory, the kinematics of QM reside in its probability structure.

If one adopts an interpretation of QM in which the state is not ontic, then I can see how the probability structure can become primary as opposed to the states. I'm just not sure I would call that "kinematics".

 

[vanhees71]

Which is dynamics, not kinematics.Which is dynamics, not kinematics.

If you are going to exclude unitary evolution from kinematics (which, as I said, I can see would make sense), then you have to also exclude any other kind of "evolution" of the state from kinematics.

This is pretty much semantics. The description of "states and observables" in Hilbert space are "kinematics" (as in classical mechanics the Euclidean space and the 1D time, the trajectories of point particles etc.) and the equations defining unitary time evolution (e.g., in the wave-mechanical formulation the time-dependent Schrödinger equation) is "dynamics" (as in classical mechanics is Newton's 2nd Law).

 

[Kolmo]

If you are going to exclude unitary evolution from kinematics (which, as I said, I can see would make sense), then you have to also exclude any other kind of "evolution" of the state from kinematics

Unitary evolution involves postulating the actual Hamiltonian and thus the interaction terms and coupling constants and how systems interact with each other over time. Bayesian updating is just a common statistical procedure whose form is completely fixed by the kinematical structure.
The view of unitary evolution as kinematical must be a very uncommon one, are there worked examples where kinematics fixes the Hamiltonian in a non-trivial manner?

 

[vanhees71]

Indeed, as in Newtonian mechanics the forces, in QT the Hamiltonians are not fixed by the "kinematics" of the theory.

 

[PeterDonis]

This is pretty much semantics.

If it is (and I'm not saying it isn't), then so are the words "kinematics" and "dynamics". They add nothing to the actual physics; they're just labels that some people like to put on certain parts of the physics.

 

[PeterDonis]

Unitary evolution involves postulating the actual Hamiltonian and thus the interaction terms and coupling constants and how systems interact with each other over time.

Yes. But this is more than just the "form" of the procedure. See below.

Bayesian updating is just a common statistical procedure whose form is completely fixed by the kinematical structure.

The "form" of Bayesian updating may be fixed by the "kinematical structure", but so is the "form" of unitary evolution: that "form" is just $i \hbar \partial_t \psi = \hat{H} \psi$, which is the same no matter what $\hat{H}$ actually is. So if the "form" is what "kinematics" is, then the "form" of unitary evolution is just as much kinematics as the "form" of Bayesian updating.

Of course, to actually do a Bayesian update, you need to know more than just the "form"; you need to know the actual priors and the actual conditional probabilities for your particular problem. Just as you need to know the actual Hamiltonian in order to actually do unitary evolution. So I do not see any difference between the two cases.

The view of unitary evolution as kinematical must be a very uncommon one

As above, one can consider just the "form" of unitary evolution to be "kinematical", in exact analogy to your statement about Bayesian updating.

 

[PeterDonis]

as in Newtonian mechanics the forces, in QT the Hamiltonians are not fixed by the "kinematics" of the theory.

But the equations those things appear in are fixed. See my post #287 just now.

 

[Kolmo]

This is a really strange view to me and I've never really heard views like yours. The kinematic/dynamic division is a very common one, calling it just semantics doesn't match anything I've read, but I'll just leave it at that.

The original point was that quantum theory is a probability theory. That seems to be borne out by the fact that:

• It belongs to a braided monoidal category like all probability theories
• It's commutative version is literally Kolmogorov probability theory
• The quantum states obeys all the usual theorems of classical probability theory such a de Finetti theorem, form of conditioning, embedding of subalgebras and hundreds of others
• It can be derived from conditions placed on families of random variables.

I don't see what suggests that the mathematical structure of quantum theory is not a probability theory.

The "form" of Bayesian updating may be fixed by the "kinematical structure", but so is the "form" of unitary evolution

How would the kinematics tell you if the evolution was Unitary or CPTP?

 

[PeterDonis]

The original point was that quantum theory is a probability theory.

The full theory of QM is, yes, because, as I've already said a couple of times now, the full theory includes measurements, which is where probabilities and discontinuities in the state enter in.

I have no problem with the above. My only issue is with using the term "kinematics" to describe the above. To me, the way QM treats measurement is dynamics, not kinematics, and, as I said, it's a weird dynamics since it involves discontinuous jumps. The only QM interpretation in which there are no such jumps in the dynamics is the MWI, and the MWI is pure unitary evolution, all the time. And you have already insisted that unitary evolution is dynamics, not kinematics. It seems very strange to me to say that and yet to still say that the full theory of QM, which is unitary evolution plus all the weird stuff happening with measurements, is "kinematics".

How would the kinematics tell you if the evolution was Unitary or CPTP?

What is CPTP?

 

[Kolmo]

My only issue is with using the term "kinematics" to describe the above

The details of the dynamics involve physical postulates that move beyond just generalized probability theory, so usually one cannot say that part is just a probability theory.

And you have already insisted that unitary evolution is dynamics, not kinematics. It seems very strange to me to say that and yet to still say that the full theory of QM, which is unitary evolution plus all the weird stuff happening with measurements, is "kinematics"

I never said that. I said the main kinematical part, i.e. not including the details of dynamical evolution, is just a generalized probability theory mathematically. Since I was claiming there was a kinematic/dynamic distinction I certainly wasn't saying "all of QM is kinematics" or anything like it.

To me, the way QM treats measurement is dynamics, not kinematics, and, as I said, it's a weird dynamics since it involves discontinuous jumps

That's the way any stochastic theory works though. For example the Black-Scholes or any stochastic process has continuous evolution and then Bayesian updating "jumps". I don't see what is weird about it.

What is CPTP?

I don't see the point in discussing QM without knowing such basic terminology. It means Completely Positive Trace Preserving. It's the general form of time evolution in QM, unitary evolution being a special case.

 

[PeterDonis]

I don't see the point in discussing QM without knowing such basic terminology.

If it's "such basic terminology", then you should be able to point me to where in all of the standard QM textbooks this term appears. For example, where is it in Ballentine?

 

[PeterDonis]

I said the main kinematical part, i.e. not including the details of dynamical evolution, is just a generalized probability theory mathematically.

But the only place probabilities arise in QM is in the context of measurement, so it seems strange to me to ignore the fundamentally dynamic nature of measurement when talking about probabilities in QM.

That's the way any stochastic theory works though.

I know that. But other stochastic theories do not claim that the system being described is actually making discontinuous jumps; the discontinuities are only in our knowledge of the system, which discontinuously changes when we obtain new data and make a Bayesian update. The system itself is assumed to have an underlying dynamics which is continuous; we just aren't able to track it precisely.

In QM, however, at least under certain interpretations, the system itself is claimed to actually discontinuously change its state when a measurement happens. That is the "weird dynamics" I am talking about.

Possibly you are implicitly using an interpretation of QM where this issue does not arise, such as the statistical or ensemble interpretation.

 

[Kolmo]

But the only place probabilities arise in QM is in the context of measurement, so it seems strange to me to ignore the fundamentally dynamic nature of measurement when talking about probabilities in QM

Probabilities arise entirely at the kinematic level in QM. It's a fact that given the algebra of observables there are no non-dispersive states. It might seem weird to you, but you don't need the dynamics to develop the probabilistic side of the theory. It's directly induced from the kinematical side.

Take classical probability theory. There we have a triplet $(\Omega, \Sigma, \mu)$, the sample space, the sigma algebra and the probability measure. We also have random variables $X: \Omega \rightarrow \mathbb{R}$. This already has all the probability structure in it. We don't need to understand the detailed dynamics of how an individual $X$ is measured to make this claim.

 

[Kolmo]

Classical mechanics is supposed to be a physical theory, not simplectic geometry

I don't really see the opposition. The fact that the systems are described by symplectic geometry is just a say very terse encoding of certain physical insights. I would see the fact that a quantum system as a whole does not obey Kolmogorov's axioms in a similar light.
Similarly axiomatisations improve my understanding, but you see them as separate to proper conceptual understanding.

 

[PeterDonis]

It's a fact that given the algebra of observables there are no non-dispersive states.

Can you give a reference that develops this in more detail? If one has already been given earlier in the thread, just point me at the post.

Also, I'd be interested in your response to this from me:

Possibly you are implicitly using an interpretation of QM where this issue does not arise, such as the statistical or ensemble interpretation.

 

[Kolmo]

Can you give a reference that develops this in more detail?

It's a corollary of Gleason's theorem. Most textbook proofs of Gleason's theorem will mention it.

 

[PeterDonis]

It's a corollary of Gleason's theorem.

Ok. I have a todo item on my list to refresh my understanding of Gleason's Theorem anyway.

 

[Kolmo]

Ok. I have a todo item on my list to refresh my understanding of Gleason's Theorem anyway.

I'd definitely have a look at the POVM based proof first:
https://arxiv.org/abs/quant-ph/9909073

It's the extension to PVMs where the real difficulty arises and requires most of the heavy duty mathematics.

 

[Kolmo]

Possibly you are implicitly using an interpretation of QM where this issue does not arise, such as the statistical or ensemble interpretation

I would say I don't see why it matters since all of this stuff has a direct analogue in classical probability theory that doesn't prevent us from using the phrase "probability theory" there.

When we set up a Stochastic model in classical probability having the usual $(\Omega, \Sigma, \mu)$ and apply it to say a dice. We wouldn't usually consider the fact that somebody could consider $\mu$ to be a real physical wave (MWI analogue) or that accounts of how a given $X: \Omega \rightarrow \mathbb{R}$, such as "Is the dice result even?", is measured would require detailed dynamics to be reasons to not call the structure $(\Omega, \Sigma, \mu)$ a probability model.

 

[PeterDonis]

all of this stuff has a direct analogue in classical probability theory

All of the stuff you have discussed does. But not all of the stuff I have discussed does; for QM interpretations that treat the quantum state as the actual, physical state of an individual system, the discontinous "jumps" in state when a measurement happens do not have a direct analogue in classical probability theory. Classical probability theory, at least as it is applied in physics, says that such "jumps" are a matter of a change in knowledge about the system, not a change in the actual state of the system. In the case of the dice, for example, nobody claims that dice undergo a discontinous change in state when we toss them. The discontinuous change is just in our knowledge of what the result of the toss is.

 

[Kolmo]

In the case of the dice, for example, nobody claims that dice undergo a discontinous change in state when we toss them. The discontinuous change is just in our knowledge of what the result of the toss is.

My point is that one could just assume the measure $\mu$ is a physical "wave" and then these discontinuous "jumps" would be just as much of a problem.

Consider for a example a classical particle in 1D with a single position variable $x$, a measurement of which is described by a single stochastic process $X(t)$. We have the probability distribution $\rho(x)$ for the position of the particle and we know $\rho \in \mathcal{L}^{1}(\mathbb{R})$. The stochastic evolution can just be formulated as a continuous map $S: t \rightarrow \mathcal{L}^{1}(\mathbb{R})$ with the state evolving over time.

Since point measures are not elements of $\mathcal{L}^{1}(\mathbb{R})$, the particle being at a specific position is not really an element of the model. So actual positions of the particle are outside the model. All we have is the distribution $\rho$ which evolves under the stochastic evolution laws, often some PDE and then we have some discontinuous "jump" upon measurements.

The only state you have obeys smooth evolution and then jumps. If one considered $\rho$ a real wave this would be an issue just as much as in quantum theory. There's nothing new here in the QM case. And none of this stuff stops us from calling this a probability model.

 

[Nullstein]

In principle, yes. But when we compare measurements at different times, then we must take into account the effect of "wave function collapse" (or state update, or whatever one likes to call it). This effect is awkward to take into account in the Heisenberg picture.

We can just compute correlation functions $\left<A(t_j)A(t_i)\right>$ ($t_j > t_i$) and see if they are compatible with a classical probability measure. For instance, if the spectrum of $A$ is bounded (as we expect for a pointer observable), we can squeeze $A$ into the interval $[-1,1]$ and check whether the (squeezed) correlators violate the CHSH inequality. The correlators are predictions of quantum theory, so all interpretations must agree on them. In this sense, this is an interpretation-free check of the Kolmogorov axioms.

$$A(t) \equiv \frac{V^{\dagger}(t)AV(t)}{\langle\psi_0|V^{\dagger}(t)V(t)|\psi_0\rangle}$$
The last equation is naturally interpreted as evolution in the Heisenberg picture, with the effect of measurement taken into account. Due to the measurement, the evolution of the observable is non-unitary (because $V(t)$ is non-unitary), random (because $\pi$ is random) and non-linear (because of the denominator).

I'm not talking about any intermediate collapse. We just use the definition $A(t) = U^\dagger(t) A U(t)$ and compute correlators.

Just to explain a bit more: Suppost we have the observable $B$ whose spectrum is contained in $[a,b]$. We define $A = 2\frac{B-a}{b-a}-1$ and $A(t)$ as explained. We choose $t_4 > t_3 > t_2 > t_1$. Then we check whether
$$\left|\left<A(t_2)A(t_1)\right>+\left<A(t_3)A(t_1)\right>+\left<A(t_4)A(t_2)\right>-\left<A(t_4)A(t_3)\right>\right|\leq 2$$
If that inequality is violated, it shows that the not all $A(t)$ can live on a joint Kolmogorovian probability space.

 

[kith]

When we set up a Stochastic model in classical probability having the usual $(\Omega, \Sigma, \mu)$ and apply it to say a dice. We wouldn't usually consider the fact that somebody could consider $\mu$ to be a real physical wave (MWI analogue) or that accounts of how a given $X: \Omega \rightarrow \mathbb{R}$, such as "Is the dice result even?", is measured would require detailed dynamics to be reasons to not call the structure $(\Omega, \Sigma, \mu)$ a probability model.

We don't consider these facts explicitly but classical stochastic models are implicitly understood as effective descriptions for situations of incomplete knowledge of the parameters of some known, more fundamental equations. Quantum theory is one of our fundamental theories so the notion that quantum theory is on par with these effective descriptions is strange.

Yes, mathematically speaking it is a kind of probability theory, but the role of these probabilities isn't understood in the same way as in all classical stochastic processes. One can declare that this isn't a problem (and -as you note- one can use quantum interpretations to reinterpret classical stochastic models) but I don't see how one can deny that there's something qualitatively different here.

(One can of course supplement quantum theory with something more fundamental but then we are talking about specific interpretations or possible extensions and not about vanilla quantum theory anymore)

 

[Kolmo]

I don't see how one can deny that there's something qualitatively different here

I'm certainly not arguing there is no qualitative difference, in fact earlier in the thread I was specifically addressing one so major it allows it to violate CHSH inequalities. I have only been saying:

• Quantum Theory is a probability theory, more general than Kolmogorov probability theory. This is a fact about its mathematical structure.
• A quantum system as a whole does not obey Kolmogorov's axioms. Which allows it to violate the CHSH inequalities.

We don't consider these facts explicitly but classical stochastic models are understood as effective descriptions for situations of incomplete knowledge of the parameters some known, more fundamental equations underlaying the model

Some people view it that way, others actually don't. For example the French probabilist Bodiou or Bruno de Finetti. You don't have to consider there to be determinisitc equations "under" Kolmogorov stochastic processes. In fact in #302 I mentioned how such deterministic states aren't even in the state space of certain stochastic processes.

 

[kith]

I'm certainly not arguing there is no qualitative difference, in fact earlier in the thread I was specifically addressing one so major it allows it to violate CHSH inequalities.

Acknowledged. I'm just not sure if we are on the same page about how fundamental this difference is. Although it can be described mathematically, the important thing for me is that it toches the question of the meaning of our most fundamental experimental notions, i.e. it touches the scientific method itself which is the most fundamental thing in physics.

I have only been saying:

• Quantum Theory is a probability theory, more general than Kolmogorov probability theory. This is a fact about its mathematical structure.

It doesn't make sense to me to say that a physical theory is some kind of mathematical theory.

Some people view it that way, others actually don't. For example the French probabilist Bodiou or Bruno de Finetti.

Do you have a reading recommendation?

You don't have to consider there to be determinisitc equations "under" Kolmogorov stochastic processes. In fact in #302 I mentioned how such deterministic states aren't even in the state space of certain stochastic processes.

If classical physics wasn't known, this would be comparable. But classical physics is known, so even if an effective description doesn't show all features of the fundamental theories there's always the straightforward possibility to see this as a shortcoming of the model (which might as well lead to testable differences).

 

[Kolmo]

Acknowledged. I'm just not sure if we are on the same page about how fundamental this difference is

It's a very very fundamental difference and not a trivial one. I might just locate it in a different place to you.

It doesn't make sense to me say that a physical theory is some kind of mathematical theory.

I just mean it in a short colloquial sense: "EM is a theory of two vector fields $E$ and $B$". It's not some sort of philosophical statement. I mean the mathematical structure of QM is that of a generalised probability theory.

Do you have a reading recommendation?

de Finetti's writings like "Probabilism" describe it well and it's covered in his two volume monograph on Probability theory.
A short account though is Sandy Zabell's article "De Finetti, Chance, Quantum Physics". It describes how de Finetti didn't view the shift to quantum theory as that surprising. There are more conceptual issues in classical probability than people often appreciate.

If classical physics wasn't known, this would be comparable. But classical physics is known, so even if an effective description doesn't show all features of the fundamental theories there's always the straightforward possibility to see this as a shortcoming of the model (which might as well lead to testable differences).

I don't think it's that simple. Just look at the difficulty with getting statistical mechanics out of something like the ergodic hypothesis. It's very hard to justify many stochastic processes in terms of underlying deterministic dynamics.

Again in my example in #302 deterministic states lie outside the state space of the stochastic process and thus are just a "hidden variable theory". For many stochastic processes, say Black-Scholes for options the resultant "hidden variable theory" is almost unimaginably remote from the phenomena and it's difficult to argue it would have scientific content.

Again this isn't meant to convey that QM and classical probability are the same, more that classical probability is not as simple as you'd think.

 

[PeterDonis]

My point is that one could just assume the measure $\mu$ is a physical "wave" and then these discontinuous "jumps" would be just as much of a problem.

Indeed it would. But nobody does that in classical physics. It is only done in quantum physics.

 

[Kolmo]

Indeed it would. But nobody does that in classical physics. It is only done in quantum physics.

I don't see why that would stop us from saying quantum theory is a probability theory though. Mathematically it is structured as a probability theory, having the same categorical structure and so forth. Surely that is enough to permit the phrase. All the arguments to the contrary apply to the classical case as well, regardless of the frequency of people who make the argument in that case.

Although you might be surprised to learn that there were views that took the classical probability as a real thing, such as in modal and propensity views of probability.

classical physics

My focus was more on classical probability theory. I'm not assuming the stochastic processes are drawn from physics. See the response to kith above. The classical probabilistic case is not as simple as this.

Consider Black-Scholes as above and de Finetti's views. It's not just "obvious" that classical probability is simply ignorance of some deterministic process, especially when deterministic states lie outside the state space and so classical distributions would not be mixtures of deterministic states and hence could not really be read as ignorance there of easily.

 

[Fra]

I agree with what Demystifier said before that it's interesting to see all the views meet. Just as it's easy to loose the numerical grip with too much philosophy, I find it's just easy to also loose the conceptual perspective and in axiomatic systems, unless the choice of axioms are extremely well founded/chosen.

One question for those who seem to represent the axiomatic road here!

The idea, to look for a theory for rational quantiative reasoning incomplete information (a "probability theory" in a very historical informal sense), in a "optimal way", is extremely symphatetic to me. But one issue I have with most of these schemes I haves seen published is that it strikes me that its still not an optimal intrisic theory of inference. As far as I can judge (which may not mean much of cours) all points towards that QM is TOO optimal. Ie. any real agent, can not represent degrees of belief with infinite precision. QM comes out as an theory of inference that is valid for when an agent with not storage or capacity constraints, are to make an "optimal inference" about a finite system.

This may seem like a practical matter, but only until you consider two such agents interacting, then exactly how "optimal" they infere, should affect their interactions.

When gravity enters the picture, and we are questioning the explanation of the hamiltonians as well, it seems that what we need, or not to axiomatise QM? Because QM may need revision anyway? What does this suggest about the axiomatic program?

I'm curious to here what the experts from that field would think about this. I have yet to find much food in the papers I have seen. But then it's a full time job to read all papers. And even worse when you have a regular job on the side. I wish I had more time to digest papers!

The original point was that quantum theory is a probability theory.

But is it the right (intrinsic) probability thery we need to make progress into QG and unification? What I am suggesting is what is an "optimal inference" also depends nthe constraints on the agent?

/Fredrik

 

[Kolmo]

But is it the right (intrinsic) probability thery we need to make progress into QG and unification?

I have no idea what is needed to make progress in QG.

 

[kith]

Again this isn't meant to convey that QM and classical probability are the same, more that classical probability is not as simple as you'd think.

I'm actually quite sympathetic to the view that we can learn a lot about notions like probability and irreversibility in classical settings by reexamining the things on which QM shines a spotlight. I guess the main difference in our views is that my thinking is grounded in physical concepts which change their shape under different interpretations while you are focused on the probabilistic nature of the mathematical structure. The downside of my view is that some of my starting points might be simplistic, the downside of your view, I think, is that you are implicitly assuming a certain interpretative framework which isn't shared by all interpretations. But I'm not entirely sure how much of this is just my dislike to call QM itself a probability theory ;-) In any case, thanks for the discussion and thanks for the reading recommendations.

 

[Kolmo]

In any case, thanks for the discussion and thanks for the reading recommendations

No problem. If you ever want to really look at this stuff D'Ariano et al's book "Quantum Theory from First Principles: An Informational Approach" derives quantum theory as a specific probability theory in explicit detail. It's quite a time commitment though. Just something to mention.

 

[kith]

If you ever want to really look at this stuff D'Ariano et al's book "Quantum Theory from First Principles: An Informational Approach" derives quantum theory as a specific probability theory in explicit detail. It's quite a time commitment though.

Thanks. I won't find the time to dive into it that deeply in the foreseeable future but a quick question if you don't mind: Is this related to Hardy's attempts at reconstructing QM? Does it deal with infinite-dimensional Hilbert spaces? (IIRC correctly, Hardy only dealt with finite-dimensional ones)

 

[kith]

If it's "such basic terminology", then you should be able to point me to where in all of the standard QM textbooks this term appears. For example, where is it in Ballentine?

As you are aware, completely positivite maps are not mentioned in many texts on general QM, but they are important in many foundations-adjacent fields like quantum information, open quantum dynamics and quantum measurement theory.

See chapter 8.2 in Nielsen & Chuang, Breuer's and Petruccione's "Theory of Open Quantum Systems" and Busch et al.'s "Quantum Measurement", for example. They might also be mentioned in the newest edition of Ballentine where he added a chapter on quantum information. I don't have access to this edition, though.