Assumptions of the Bell theorem

[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

 

[Demystifier]

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

I looked at the second paper. I was hoping to see something like a generalization of Kolmogorov axioms, but I was not able to recognize anything like that. Any hint?

 

[Fra]

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

Can we try to convery what a generalized probability theory is, conceptually, in nontechnical sense? That's not to say details aren't important, but just to increase understanding between the extremal perspectives?

Can we think of generalised probability theory (which I THINK is effectivelty the same as what I tend to call a theory of inference) as something like this

- There are some kind of structures, or states, living in some space spaces, that represents how information is ENCODED
- Composition rulels for how to define logical junctions or combinations or two spaces
- Some set of transformations that mapes one state from one space into another state of another space (loosely thought of as recoding information)
- And most importantly a way to compute a measure of degree of belieif from a given set or junction of states in several spaces? This correspons to how to compute teh expectations from the states.

Except of course a mathematician will make all this detailed and less comprehensible.
If you agree with this, I think even for non mathematical emphaiss can easily see in what sense existing probability theories are special cases.

I think it¨s also easy to see how further questions related to this reconstruction. For example, shall we make ontological associations of state spaces and agents internal structure or not? WHO is supposed to make the above "computations" and physical endode the states? agents? or is it just publicly encoded in the whole universe?

/Fredrik

 

[Kolmo]

I looked at the second paper. I was hoping to see something like a generalization of Kolmogorov axioms, but I was not able to recognize anything like that. Any hint?

The whole of section II lays out the formalism and concepts of general probability theories.

 

[vanhees71]

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?

No, quantum theory is not deterministic to begin with. I've only said that the time evolution is unitary for closed systems but not for open ones, and the issue related to the "measurement problem" is not about closed but open systems, and considering it as such, there is no more "measurement problem" related to unitary time evolution.

 

[Kolmo]

No, quantum theory is not deterministic to begin with. I've only said that the time evolution is unitary for closed systems but not for open ones

Just to add as a note of interest in QFT all systems are open (Reeh-Schlieder theorem).

considering it as such, there is no more "measurement problem" related to unitary time evolution

I always thought Julian Schwinger's remarks about this were good. Basically he said the measurement problem only really arises if you take the closed system formalism in von Neumann's book (which has some slight errors anyway) and model the measuring device as a two-level system/qubit (which von Neumann does). It's not surprising that such a simple unrealistic model has problems.

If one looks at more detailed treatments like Allahverdyan et al's 2013 Phys. Rep. paper "Understanding quantum measurement from the solution of dynamical models" more detailed treatments handle the measurement process well.

 

[Demystifier]

No, quantum theory is not deterministic to begin with. I've only said that the time evolution is unitary for closed systems but not for open ones, and the issue related to the "measurement problem" is not about closed but open systems, and considering it as such, there is no more "measurement problem" related to unitary time evolution.

So in the following let us consider only closed systems. How does the probabilistic interpretation work for closed systems? More specifically: Is the Born rule valid? Does its validity depend on the existence of measurement? If the Born rule is valid even without measurement, then what determines the basis of the Born rule? Or if measurement is essential for validity of the Born rule, then what is the difference between measurement and non-measurement in a closed system? Last but not least, are all these questions even relevant, or are they just empty philosophy?

 

[Demystifier]

Just to add as a note of interest in QFT all systems are open (Reeh-Schlieder theorem).

Except for the whole universe.

 

[Demystifier]

I always thought Julian Schwinger's remarks about this were good. Basically he said the measurement problem only really arises if you take the closed system formalism in von Neumann's book (which has some slight errors anyway) and model the measuring device as a two-level system/qubit (which von Neumann does). It's not surprising that such a simple unrealistic model has problems.

Where exactly did Schwinger say that?

 

[Kolmo]

Except for the whole universe.

I don't really know if what you are getting at has a proper meaning in QFT. The "whole universe" here would have to be considered to be something like the global algebra $\mathcal{A} = \overline{\cup_{\mathcal{O}} \mathcal{A}(\mathcal{O})}$ with $\mathcal{A}(\mathcal{O})$ the algebra of each region. Only such a global algebra would have the PVMs required to discuss some type of global pure state that would be "closed" in the usual sense in QM.

However there is a difficulty with forming such an algebra in theories with massless modes and general spacetime curvature and regardless the entire theory can be handled with just the local algebras anyway, so I don't see any issues stemming from some global pure state.

 

[Demystifier]

I don't really know if what you are getting at has a proper meaning in QFT. The "whole universe" here would have to be considered to be something like the global algebra $\mathcal{A} = \cup_{\mathcal{O}} \mathcal{A}(\mathcal{O})$ with $\mathcal{A}(\mathcal{O})$ the algebra of each region. Only such a global algebra would have the PVMs required to discuss some type of global pure state that would be "closed" in the usual sense in QM.

However there is a difficulty with forming such an algebra in theories with massless modes and general spacetime curvature and regardless the entire theory can be handled with just the local algebras anyway, so I don't see any issues stemming from some global pure state.

It seems that you are saying that QFT is not even defined for the whole Universe, but only for its subsystems. So however big the subsystem is, there is always something out of it which can be considered as a "measuring apparatus" for the subsystem. Is that right?

 

[Kolmo]

It seems that you are saying that QFT is not even defined for the whole Universe, but only for its subsystems. So however big the subsystem is, there is always something out of it which can be considered as a "measuring apparatus" for the subsystem. Is that right?

There's a von Neumann algebra of observables for each individual region but not for the universe as a whole. Even when you try to define it you need special conditions and even if you can define it when you have massless modes it only has mixed states.

 

[Demystifier]

There's a von Neumann algebra of observables for each individual region but not for the universe as a whole. Even when you try to define it you need special conditions and even if you can define it when you have massless modes it only has mixed states.

Do you know how all this works in curved spacetime when the universe is closed and hence finite?

 

[Kolmo]

Do you know how all this works in curved spacetime when the universe is closed and hence finite?

I'm not sure of the relevance or what you would expect to change. Certainly the formulation works in such a case.

 

[Demystifier]

I'm not sure of the relevance or what you would expect to change. Certainly the formulation works in such a case.

I mean when the universe is finite (say with a spherical or toroidal global topology), perhaps then there is no problem of having algebra of operators for the whole universe.

 

[vanhees71]

Just to add as a note of interest in QFT all systems are open (Reeh-Schlieder theorem).I always thought Julian Schwinger's remarks about this were good. Basically he said the measurement problem only really arises if you take the closed system formalism in von Neumann's book (which has some slight errors anyway) and model the measuring device as a two-level system/qubit (which von Neumann does). It's not surprising that such a simple unrealistic model has problems.

If one looks at more detailed treatments like Allahverdyan et al's 2013 Phys. Rep. paper "Understanding quantum measurement from the solution of dynamical models" more detailed treatments handle the measurement process well.

I also highly recommend to read the "Prologue" in Schwinger's QT textbook (otherwise, it's only something for somebody very advanced with the theory ;-)):

J. Schwinger, Quantum Mechanics, Symbolism of Atomic
Measurements, Springer, Berlin, Heidelberg, New York (2001).

 

[vanhees71]

So in the following let us consider only closed systems. How does the probabilistic interpretation work for closed systems? More specifically: Is the Born rule valid? Does its validity depend on the existence of measurement? If the Born rule is valid even without measurement, then what determines the basis of the Born rule? Or if measurement is essential for validity of the Born rule, then what is the difference between measurement and non-measurement in a closed system? Last but not least, are all these questions even relevant, or are they just empty philosophy?

You cannot discuss the measurement problem by considering only closed systems.

I also thought that we all agree on what the physical, i.e., minimally interpreted, formalism is about: You have a Hilbert space, states (positive semidefinite self-adjoint trace-one operator) and an observable algebra represented by (essentially) self-adjoint operators. The possible values of observables are given by the spectrum of its representing self-adjoint operator. If a system is prepared in a state $\hat{\rho}$ it implies probabilities and only probabilities for the outcome of measurements. If $a$ is an eigenvalue of the corresponding representing operator $\hat{A}$ and $|a,\alpha \rangle$ a complete set of (generalized) eigenstates to the eigenvalue $a$, then the probability to find the value $a$ is given by $P(a)=\sum_{\alpha} \langle a,\alpha|\hat{\rho}|a,\alpha \rangle$. Of course you have to formulate this out in more detail to get also the "continuous eigenvalues" mathematically correct, but that's not the point here.

Physicswise measurements obviously exist, because QT applied in this physical minimal interpretation simply works, i.e., it correctly predicts the measured phenomena. For me everything else is indeed just "empty philosophy".

 

[stevendaryl]

I always thought Julian Schwinger's remarks about this were good. Basically he said the measurement problem only really arises if you take the closed system formalism in von Neumann's book (which has some slight errors anyway) and model the measuring device as a two-level system/qubit (which von Neumann does). It's not surprising that such a simple unrealistic model has problems.

I don't see that, at all.

My take on the Born rule as it is relevant to measurements is this:

We have a microsystem (a small number of particles interacting). We can describe its state at a given time by a vector in a hilbert space $|\psi\rangle$. (Or maybe a density matrix, but let me stick to pure states for simplicity, unless the difference turns out to be important.)

Then we have a measuring device. The state of the measuring device is typically described in macroscopic terms, you know, the orientation of the device, the presence/absence of spots on a photographic plate, etc.

Then we use a hybrid of classical and quantum reasoning to describe the interaction of the device with the microsystem. We establish a transition rule of the form

$D_i \times |\psi_\lambda\rangle \Longrightarrow D_\lambda$

where $D_i$ is the initial state of the device, $|\psi_\lambda\rangle$ is some collection of orthonormal states in the Hilbert space of the microsystem, and $D_\lambda$ is a collection of macroscopically distinguishable final states of the device. The meaning of this transition rule is: "If the device starts off in its initial state, and the microsystem is in state $|\psi_\lambda\rangle$, then the interaction of the two will reliably result in the device making a transition to the corresponding final state $D_\lambda$"

(There's a lot of fuzziness here, such as: what does "reliably" mean? And also, what does the "state" of a device mean. For our purposes, the $D_i$ and $D_\lambda$ are just descriptions, rather than complete specifications. The description might be something like "This LED is turned on" or "There is a spot on the left side of the photographic plate.)

Note, that our description of the transition rule is only for specific states $|\psi_\lambda\rangle$. The rule doesn't say anything about how the device behaves if the microsystem is in any other state. That's where Born's rule comes into play. Given the above transition rules for the microstates $|\psi_\lambda\rangle$, Born's rule implies the following transition rule for a superposition of states:

$D_i \times \sum_\lambda c_\lambda |\psi_\lambda\rangle \Longrightarrow D_\lambda$ with probability $|c_\lambda|^2$

 

[Kolmo]

I also highly recommend to read the "Prologue" in Schwinger's QT textbook (otherwise, it's only something for somebody very advanced with the theory ;-)):

J. Schwinger, Quantum Mechanics, Symbolism of Atomic
Measurements, Springer, Berlin, Heidelberg, New York (2001).

It's a phenomenal book, a long time favourite of mine. Such clear exposition.

It informs the approach in BE Englert's own introductory QM books, using a series of Stern-Gerlach devices to derive the presence of complex numbers, the bra-ket formalism and so on. A series of three small volumes, very useful for an introduction to quantum theory.

 

[Kolmo]

I don't see that, at all.

My take on the Born rule as it is relevant to measurements is this:

This seems simple enough, but I don't really get what the issue is.
As far as I can see from studying things like the Allahverdyan paper and other more detailed treatments, the only real issues such as the absence of conditionalization are covered by more detailed models.

 

[Demystifier]

You cannot discuss the measurement problem by considering only closed systems.

Physicswise measurements obviously exist, because QT applied in this physical minimal interpretation simply works, i.e., it correctly predicts the measured phenomena. For me everything else is indeed just "empty philosophy".

Are you saying that closed systems are empty philosophy? That a quantum system can only be understood if a part of the total system is left un-analyzed?

And at the same time, you say this un-analyzed part of the system is described by quantum theory too?

So you essentially say that the correct way to analyze the measuring apparatus by quantum theory is to not analyze the measuring apparatus by quantum theory?