Assumptions of the Bell theorem

[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.

 

[PeterDonis]

They might also be mentioned in the newest edition of Ballentine where he added a chapter on quantum information.

I have that edition and I don't remember seeing the term there. But I don't have my copy handy at the moment to check.

 

[Demystifier]

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

The point is, if you don't take into account the effect of measurement (intermediate collapse), then the correlator you compute does not correspond to the measured correlation. See e.g. https://arxiv.org/abs/1610.03161

 

[Kolmo]

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)

It is, the field has moved on a bit since then. There are now at least eight constructions of quantum theory. These don't share the weaknesses of Hardy's which contained certain bounding statements, by which I mean Hardy gave axioms that came with conditions like "QM is the theory obeying this for which the parameter $N$ is smallest..." rather than axioms that just directly fix QM.

D'Ariano's axioms can give infinite-dimensional QM, he has given modifications to obtain certain QFTs. The textbook I mentioned doesn't go into the infinite-dimensional case since it's for undergraduates.

the downside of your view, I think, is that you are implicitly assuming a certain interpretative framework which isn't shared by all interpretations

I would say two things to this having thought about it:

• My main point has been I don't see why this matters for calling the theory a probability theory. There are actually several philosophical views of General Relativity, but it would be odd to use that to claim General Relativity is not a geometric theory of gravity since mathematically it uses differential geometry.
• I don't really see a practical downside here honestly. Take interpretive ideas to move beyond the probabilistic framework for quantum theory such as hidden variable theories. Has anybody ever used such ideas to compute the Lamb shift for example?

I guess what would help me would be if I could see say an explanation of two features of quantum theory using one of these non-probabilistic approaches:

• So say how does one explain that the quantum state obeys a de Finetti representation theorem? What does that mean in a non-probabilistic setting. It's hard for me to understand this otherwise since de Finetti's representation represents a passage from basic credences to inductive hypothesis testing. I find it hard to understand how the quantum state obeys such an "inferential" theorem if it is not understood in a probabilistic sense.
• Why is there no information in the quantum state beyond the probabilities it assigns to projectors in the operator algebra? The whole state can be reconstructed from the probability assigned to each projector, even in QFT. So there doesn't seem to any more information in the state to me.

The entire structure of the theory is mathematically a generalization of classical probability theory, with the quantum state obeying several identical theorems to the point where classical probabilistic theorems can literally just be "lifted" straight into the framework. The fully general projection postulate, i.e. Lüder's as opposed to von Neumann's erroneous one in his textbook, is literally the exact same structure as Bayesian conditioning, i.e. a unit norm projection onto a subalgebra.

I still don't get why in light of all these hundreds of probabilistic theorems proven for the formalism, proofs that the formalism is an element of the general set of probability theories that one should somehow avoid saying it is a probability theory.

 

[Demystifier]

I still don't get why in light of all these hundreds of probabilistic theorems proven for the formalism, proofs that the formalism is an element of the general set of probability theories that one should somehow avoid saying it is a probability theory.

Note that the so called "quantum probability theory" is usually not studied in mathematical textbooks on probability theory. Indeed, quantum probability can still be interpreted in terms of Kolmogorov probability, by using many models of Kolmogorov probability depending on the context. How exactly the change of context changes the model is a perhaps a central question, but this is a physical question, not merely a question in the theory of probability.

 

[vanhees71]

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

Indeed, and you can say that "kinematics" is the part of the math that enables you to write down these equations. The concrete choice of the forces (or rather the Hamiltonian or action) is "dynamics", but as I said above, it's pure semantics. It doesn't add much to the content of physical theory, where you draw the line between "kinematics" and "dynamics".

 

[vanhees71]

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.

One should, however, stress that in quantum theory there are no quantum jumps. The time evolution of the probabilities is governed by differential equations and are thus smooth. Quantum jumps and/or collapse are just FAPP descriptions for pretty fast transition processes due to the interaction of the investigated system with the environment/measurement device leading to decoherence and irreversible defined measurement results.

 

[Kolmo]

Indeed, quantum probability can still be interpreted in terms of Kolmogorov probability, by using many models of Kolmogorov probability depending on the context

Any General Probability Theory is Kolmogorov within a fixed context. I don't see what that changes. The whole point is that all random variables as a whole are not, that is what makes them more general probability theories.

 

[vanhees71]

Indeed, in QT "all random variables" are simply "all observables" of a system, but only the probabilities for the outcome of physically feasible ideal measurements obey Kolmogorov's axioms. You cannot measure "all observables" accurately at one single system.

 

[Demystifier]

Quantum jumps and/or collapse are just FAPP descriptions for pretty fast transition processes due to the interaction of the investigated system with the environment/measurement device.

Interactions are unitary and deterministic, the results of measurement outcomes are not. So there is something which is not merely a result of interactions. Whether this something is instantaneous or continuous, we don't know. A minimal interpretation should remain agnostic on that.

 

[vanhees71]

The time evolution of closed systems is unitary (I don't know what your statement that "interactions are unitary and deterministic means). Here we talk about interactions of the system with a macroscopic measurement device, i.e., an open quantum system. The time evolution of the system alone is not unitary but governed by some (usually non-Markovian) master equation, and these can describe the measurement process and decoherence. The minimal interpretation is not agnostic on that and it shouldn't be. It only underlines the fact that hitherto there seems to be no necessity for extending QT to solve the (in my opinion only appararent) "measurement problem".

 

[Demystifier]

Any General Probability Theory is Kolmogorov within a fixed context. I don't see what that changes. The whole point is that all random variables as a whole are not, that is what makes in them more general probability theories.

One should distinguish axioms from various models that satisfy the axioms. Take, for instance, axioms of group theory and their various models like SU(2), SU(3), etc. If someone told you that in one context measurement results are described by group SU(2) while in another context measurable results are described by group SU(3), would you say that the whole theory requires a generalization of group theory axioms? I would say no, it's still the same axioms of group theory, realized by different models in different contexts. The analogy with Kolmogorov probability axioms and its models should be obvious.

 

[Demystifier]

The time evolution of closed systems is unitary (I don't know what your statement that "interactions are unitary and deterministic means). Here we talk about interactions of the system with a macroscopic measurement device, i.e., an open quantum system.

Are you saying that a closed quantum system is deterministic? Are you saying that the Born rule is not valid in a closed quantum system?

 

[Kolmo]

The analogy with Kolmogorov probability axioms and its models should be obvious

It's not.

The analogy would be that Kolmogorov theory would be like Abelian groups. The fact that groups might have Abelian subgroups doesn't mean the general theory is basically just the restricted Abelian case. It's the same with general probability theories, individual contexts might obey Kolmogorov's theory but the general structure is much more general than just Kolmogorov's case.

Kolmogorov theory is specifically measures on sigma algebras of sets, general probability theories are any strict symmetric monoidal category, of which classical probability theory and quantum operator algebras are just special cases.

 

[Demystifier]

general probability theories are any strict symmetric monoidal category

Where can I read more about general probability theories?

 

[Kolmo]

Where can I read more about general probability theories?

Monoidal categories are nice because they can be given a diagrammatic representation for which:
Selinger, P. 2011. A survey of graphical languages for monoidal categories. Pages 289–355
of: New Structures for Physics
is a nice reference.

For GPTs this paper has a nice introduction in more physicist terms:
Chiribella, G., D’Ariano, G. M., and Perinotti, P. 2010a. Probabilistic theories with purification. Phys. Rev. A, 81(6), 062348