A Assumptions of the Bell theorem

[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?

 

[vanhees71]

No, I'm saying that you can't solve the "measurement problem" by considering only closed quantum systems. This is the one thing where Bohr was right: Measuring a quantum system means to use a macroscopic apparatus to gain information about this quantum system, and a measurement implies an irreversible process letting me read off a pointer at that instrument. This you cannot describe by a closed system (neither in classical mechanics or field theory nor in quantum (field) theory).

In classical as well as quantum physics you derive the behavior of macroscopic systems by a plethora of methods leading to an effective description leading to irreversibility, dissipation and particularly, in the quantum case, decoherence.

What is empty philosophy is to claim you can describe macroscopic systems in all microscopic detail as closed quantum systems. It's also empty philosophy to claim that there's a measurement problem only because of this impossibility. If you wish it's the same empty philosophy as to claim that there is a problem, because we are able to describe nature with mathematical models at all. It's just an observed fact that we can to an amazing extent, as it is an observed fact that we are able for the last 400+x years to invent better and better instruments to measure and thus quantify with higher and higher accuracy all kinds of phenomena, and this is enabled by both these experimental and engineering achievements and its interplay with theory, which enables us to think and talk about phenomena that exceed or direct abilities by several orders of magnitude in scales (from the microscopic subatomic/subnuclear dimensions below 1 fm up to very large astronomical if not even cosmological scales). It's empty philosophy (though pretty entertaining sometimes) to ask, why this quantitative description of our universe is possible at all.

 

[stevendaryl]

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.

I don’t think so. You have deterministic evolution of microsystems, but when you include macroscopic systems (measurement devices), it becomes nondeterministic? How is that possible if the macro system is governed by the same laws as the micro system?

 

[stevendaryl]

No, I'm saying that you can't solve the "measurement problem" by considering only closed quantum systems.

Does that mean that you can’t have measurements in a closed system?

 

[Kolmo]

I don’t think so. You have deterministic evolution of microsystems, but when you include macroscopic systems (measurement devices), it becomes nondeterministic? How is that possible if the macro system is governed by the same laws as the micro system?

I don't really understand this comment. This is how I see it.

Well say we treat the relevant degrees of freedom of the device as $D_i$ following your notation. They form some commutative algebra $\mathcal{M}$. The quantum system then has some non-commutative algebra $\mathcal{A}$ so that the total system has the algebra $\mathcal{M}\lor\mathcal{A}$ or if you are in non-relativistic QM $\mathcal{M}\otimes\mathcal{A}$.

Then if we have some state $\omega$ on $\mathcal{M}\lor\mathcal{A}$ that is essentially a product state and some time evolution $\alpha_{t}$ that couples them we end up with a mixed state after measurement of the form:
$\omega = \sum_{i} p_{i}\otimes\rho_{i}$
with $p_{i}$ a state on the device's algebra and $\rho_{i}$ a state of the quantum system, potentially even a mixed one. Since $p_{i}$ is just a state on a commutative operator algebra then (by Gelfand's representation theorem if one wants to fully prove it) it's just some classical probability distribution and thus just ignorance of the current device reading. Upon seeing the device reading you condition on the outcome $i$ via Bayes's rule which has an effect on the total state such that the state for the quantum system is now $\rho_i$.

The only issue remaining would be can we in fact treat macroscopic degrees of freedom as commutative in this sense. To me this was answered in sufficient detail in the affirmative in older works by Ludwig, Gottfried's textbook or the long papers of Loinger et al from the 60s about ergodicity in quantum theory. There were other works by Peres, Omnes, Lockhart and others in the 80s that come to the same conclusion by either properties of the observable algebra in QFT, decoherence studies or properties of coarse grained observables. Even more recently you have the enormous synthasizing and summarizing papers of Allahverdyan et al.

 

[stevendaryl]

Well say we treat the relevant degrees of freedom of the device as $D_i$ following your notation. They form some commutative algebra $\mathcal{M}$. The quantum system then has some non-commutative algebra $\mathcal{A}$ so that the total system has the algebra $\mathcal{M}\lor\mathcal{A}$ or if you are in non-relativistic QM $\mathcal{M}\otimes\mathcal{A}$.

Then if we have some state $\omega$ on $\mathcal{M}\lor\mathcal{A}$ that is essentially a product state and some time evolution $\alpha_{t}$ that couples them we end up with a mixed state after measurement of the form:

But why should a product state evolve into a mixed state?

 

[Kolmo]

But why should a product state evolve into a mixed state?

I don't get this. It just does given the evolution (super)operator, I'm not sure what "why" I could provide.

 

[stevendaryl]

But why should a product state evolve into a mixed state?

Decoherence does not describe the evolution of a pure state into a mixed state. It becomes a mixed state when we trace over the environmental degrees if freedom. That’s not something that happens, that’s a mathematical choice that the analyst uses. To me, something can’t go from (1) a superposition of a state in which A is true and a state where B is true to (2) either A or B is true, we just don’t know which until we gather more information, just because we have performed a mathematical operation on a state.

 

[Kolmo]

Decoherence does not describe the evolution of a pure state into a mixed state

I didn't discuss decoherence, except tangentially at the end. The process described above doesn't involve decoherence.

 

[stevendaryl]

It’s my opinion that many people believe just contradictory things about quantum mechanics. When it comes to something simple like an electron, there is a clear difference between being in a superposition of states A and B and being in a mixed state with a probability of being in state A and a probability of being in state B. But when it comes to macroscopic objects, people ignore the distinction.

 

[stevendaryl]

I don't get this. It just does given the evolution (super)operator

I don’t think it does.