Quantum physics vs Probability theory

[Killtech]

TL;DR Summary

Why can't Quantum Physics be modeled in classical probability theory (i.e. classical Kolmogorovs definition of probability)?

Because I do have a background in the latter it was originally very difficult for me to understand some aspects of QP (quantum physics) when I initially learned it. More specifically whenever probabilities were involved I couldn’t really make full sense of it while I never had any problems understanding the actual math (i.e. the deterministic part). Basically I had big troubles of anything involving the QP interpretations and it took me a while to learn that QP uses a different concept of probability simply not compatible with PT (probability theory). I found there are rigorous formulations of this like under the keyword of quantum probability and it helped me a great deal in finding a way how to view and interpret these calculations.

But this does not answer another question: why even go this way and sway away from Kolmogorovs PT? For one I do not know of any experiment that exhibits any weird nature that would pose a problem for modelling within classic PT (non-locality isn’t an issue as PT is inherently non-local i.e. it doesn’t even have a concept of the underlying physical space and therefore its metric). That however would be a key to comprehend the core nature of QM for me. Without this it leaves a big gap in my understanding as to why it is necessary to model QP the way it is/ what necessitates such modelling. I did not find much to read on this particular topic hence this post and question.

 

[Stephen Tashi]

But this does not answer another question: why even go this way and sway away from Kolmogorovs PT?

I don't know any aspect of QM that contradicts or departs from Kolmogorov's approach to probability theory, which is based on the definition of a "probability Space" that consists of a space of outcomes, a sigma algebra of subets of those outcomes, and a probability measure defined on the sets in the sigma algebra.

In the "classical" application of probability theory, two randomly varying quantities $X,Y$ would be modeled by a joint probability distribution $F(x,y)$. There might be debate about what function $F(x,y)$ to use, but there wouldn't be any doubt that the general approach is viable. If $A$ and $B$ are sets of values, the joint probability distribution can be used to compute the probability that (simultaneously) $X \in A$ and $Y \in B$. However, as you know, in QM there are pairs of randomly varying quantities that do not simultaneously have specific values. So attempts to model such populations using a joint probability distribution fail.

You can find literature about "quantum probabilities", including claims that they correspond to classical probabilities or to Bayesian probabilities. (e.g.
https://arxiv.org/abs/1406.4886 ). As far as I can see, such discussions revolve around how the Kolmogorov approach to probability theory can or cannot be applied, not about a different approach to the fundamentals of probability theory.

 

[PeroK]

Summary:: Why can't Quantum Physics be modeled in classical probability theory (i.e. classical Kolmogorovs definition of probability)?

Because I do have a background in the latter it was originally very difficult for me to understand some aspects of QP (quantum physics) when I initially learned it. More specifically whenever probabilities were involved I couldn’t really make full sense of it while I never had any problems understanding the actual math (i.e. the deterministic part). Basically I had big troubles of anything involving the QP interpretations and it took me a while to learn that QP uses a different concept of probability simply not compatible with PT (probability theory). I found there are rigorous formulations of this like under the keyword of quantum probability and it helped me a great deal in finding a way how to view and interpret these calculations.

But this does not answer another question: why even go this way and sway away from Kolmogorovs PT? For one I do not know of any experiment that exhibits any weird nature that would pose a problem for modelling within classic PT (non-locality isn’t an issue as PT is inherently non-local i.e. it doesn’t even have a concept of the underlying physical space and therefore its metric). That however would be a key to comprehend the core nature of QM for me. Without this it leaves a big gap in my understanding as to why it is necessary to model QP the way it is/ what necessitates such modelling. I did not find much to read on this particular topic hence this post and question.

I don't really understand the issue. The wave function represents a complex probability amplitude. Not a probability. The modulus squared of the wave function represents the probability density function.

This is the way nature works, so you have no choice.

For more background on this, you could try the page on Scott Aaronson's blog.

 

[Killtech]

In the "classical" application of probability theory, two randomly varying quantities $X,Y$ would be modeled by a joint probability distribution $F(x,y)$. There might be debate about what function $F(x,y)$ to use, but there wouldn't be any doubt that the general approach is viable. If $A$ and $B$ are sets of values, the joint probability distribution can be used to compute the probability that (simultaneously) $X \in A$ and $Y \in B$. However, as you know, in QM there are pairs of randomly varying quantities that do not simultaneously have specific values. So attempts to model such populations using a joint probability distribution fail.

this is kind of the problem i am dealing with in finding my answer: you are already putting in a lot of interpretation to be able to make this statement. You need a very specific underlying sample/state space $\Omega$ to give rise to random variables that have such a property. but if you try to properly model an quantum experiment as you would any random experiment you'll find PT won't simply let you use such a state space to begin with. but that isn't an obstacle because you just have to model it somewhat differently - or at least i haven't found an experiment where this wouldn't work.

If you consider that from a PT point of view you only have to describe how the input information (experimental setup, etc) relates to the output information (measured results). In fact from this very abstract view of input to output mapping there is really nothing remotely non-classic at all, so it's seems to be actually straight forward to model.

The thing is that the state space will always reflect the all possible input x output information and you naturally start only with random variables that map input to detector output state. With that in mind how would you construct a pair of random variables that don't have specific simultaneous values in that probability space? the problem is that detector output is always sharp and so is the 'known' input information thus i don't think it's possible to do and therefore i don't see this work in order to construct a reductio-ad-absurdum kind of proof from this.

I don't really understand the issue. The wave function represents a complex probability amplitude. Not a probability. The modulus squared of the wave function represents the probability density function.

This is the way nature works, so you have no choice

No, that's how QM works, not nature. nature does not depend on how we formulate it. Okay, i admit this is not a probability theory forums so it may be a bit off topic to ask how to model QM experiments in terms of classical PT. But the question if the nature of QM has anything in store that would make it impossible - i don't know where else to ask it.

 

[PeroK]

No, that's how QM works, not nature. nature does not depend on how we formulate it. Okay, i admit this is not a probability theory forums so it may be a bit off topic to ask how to model QM experiments in terms of classical PT. But the question if the nature of QM has anything in store that would make it impossible - i don't know where else to ask it.

First, you've labelled ths as a "B" level thread, which implies you do not know undergraduate level maths or physics. Is that correct?

Technically, of course, you are correct. Probability amplitudes are part of QM. But, to take an example: how do you model the double-slit experiment (and interference in general) using real, positive classical probabilities?

Also are you aware of Bell's theorem?

 

[meopemuk]

Any probability theory should be based on a theory of logic, i.e., before assigning probabilities to questions or propositions we should specify the set of questions/propositions and basic logical operations between them (and, or, not, etc.). Behind classical probability theory there is the Aristotelean/Boolean logic whose questions/propositions are in one-to-one correspondence with subsets of a certain set. In classical mechanics, these are subsets of the phase space.

In 1936 Birkhoff and von Neumann proposed a brilliant idea that the whole point of quantum mechanics is that it is based on a different non-Boolean logic:

G. Birkhoff and J. von Neumann, "The logic of quantum mechanics", Ann. Math., 37 (1936), 823.

In this "quantum logic" the Boolean distributive law is replaced by a more general orthomodular postulate, and questions/propositions can be realized as subspaces of a certain Hilbert space.

This idea is beautiful, because in one stroke it explains all seemingly illogical paradoxes of quantum mechanics. For example, it explains why in the two-slit experiment we have to add the probability amplitudes, not the probabilities themselves. I also recommend writings of Hilary Putnam who rigorously defended the ideas of quantum logic and quantum probability in the 1960's and 1970's. See, e.g.

H. Putnam, "How to think quantum-logically", Synthese 29 (1974), 55.

and references there.

Eugene.

 

[Killtech]

First, you've labelled ths as a "B" level thread, which implies you do not know undergraduate level maths or physics. Is that correct?

Technically, of course, you are correct. Probability amplitudes are part of QM. But, to take an example: how do you model the double-slit experiment (and interference in general) using real, positive classical probabilities?

Also are you aware of Bell's theorem?

Oh, sorry in this cased i mislabeled it. The "B" was meant to characterize the question (i though it to be basic/fundamental) and not my knowledge level. i studies physics along/after finishing with math. but i work in insurance for quite some time now and didn't have much contact since - apart from my personal interest that is.

As for Bell: yeah of course. this is why i mentioned that PT is inherently non local... in a sense (it's not defined so no questions about locality is decidable in a PT model of a random experiment unless that modeling explicitly adds it). That said i don't see where one could run into problems when describing Bells experiments in terms of PT.

Lastly the double slit experiment: i actually tried to think through exactly that case in an attempt to construct a reductio-ad-absurdum scenario... but failed to produce any contradictions so far.

So my first try was simple: fixed geometry (slit width, distance from of detectors etc) - so basically input information is always the same - so it's void from a modelling point of view. And there is only the output information - i.e. the possible detector states. So the sample space is simply made of all possible detector readings and because there is no input information there isn't really anything to model. It’s just a plain mapping towards the empirical measured probability distribution and that is actually all that can be predicted for this particular case.

Okay, i realized that this attempt is not helping me at all. But it showed that PT modeling will have a very minimalist nature and won't make statements about any stuff not directly relevant for the experiment. So i tried a more complex scenario: completely free geometry i.e. positioning of walls, slits and detectors are all freely choosable and part of the input information. That way PT can't get around modeling the time evolution of the system. Now this is the point where it gets into less solid territory since there is no unique/canonical way to do this and I am not sure I get everything right. In any case I did not run into anything that would indicate this not to be possible. I could write down what I have so far.

 

[Killtech]

Any probability theory should be based on a theory of logic, i.e., before assigning probabilities to questions or propositions we should specify the set of questions/propositions and basic logical operations between them (and, or, not, etc.). Behind classical probability theory there is the Aristotelean/Boolean logic whose questions/propositions are in one-to-one correspondence with subsets of a certain set. In classical mechanics, these are subsets of the phase space.

And here is the thing: any questions or propositions one could come up with, when formulated by the means of all that is measurable/results detectors can yield stays within the realm of Boolean logic. QM experiments (not their formulation in QM) don't introduce anything new to the table. characterized only by the set of all possible measurable outcomes they don't distinguish from any other type of experiments so they remain in that classic logic realm.

However QM formulation splits the model into a deterministic time evolution and a probabilistic interpretation part. this way it opens the possibility to introduce things between the mechanics and its interpretation for which some questions need a non-boolean logic. and this is a big difference compared to classical PT: it won't allow you to split the mechanics from their interpretation in the first place. this way it will always be stuck with only what detectors can measure.

In 1936 Birkhoff and von Neumann proposed a brilliant idea that the whole point of quantum mechanics is that it is based on a different non-Boolean logic:

I do know these formulations and while they offer interesting insights and a good and consistent way to deal with the probabilistic part of QM they don't answer my simple question: is there any prove experimental or otherwise that they are necessary/the only way to formulate QM? if so, what fundamental observation/behavior is it that enforces it?

 

[meopemuk]

I do know these formulations and while they offer interesting insights and a good and consistent way to deal with the probabilistic part of QM they don't answer my simple question: is there any prove experimental or otherwise that they are necessary/the only way to formulate QM? if so, what fundamental observation/behavior is it that enforces it?

I think the most important message from Birkhoff, von Neumann and Putnam is that the logic we use in science and in math is not a universal God-given truth, but an object of experimental/theoretical studies, like all other natural phenomena. Of course, when we are dealing with well-defined entities like macroscopic bodies or geometrical figures (both of them have sets of unambiguous properties that exist simultaneously), we should apply to them the laws of Boolean logic (including the distributive law) and classical probability. That's what we do in classical physics and in math.

However, how can we be sure that the same logical laws apply to measurements of the electron's position and momentum? These measurements are inherently probabilistic and can't be done simultaneously. So, it is only natural to assume that these measurements can be ruled by a different logic.

By the way, the "probabilistic part of QM" is also a direct consequence of quantum logic, because due to Gleason's theorem, all state-like measures on (orthomodular) lattices of subspaces in a Hilbert space are necessarily probabilistic. So, there is no mystery in the split between "a deterministic time evolution and a probabilistic interpretation part."

As far as I know, mathematicians have invented a lot of various formal logical systems (in addition to the classical Boolean logic and quantum orthomodular logic). So, perhaps one of these systems could describe nature even better than the quantum logic? I don't think there is a proof that the orthomodular quantum logic is the only possibility, but so far it worked very well and I don't see any need to change it.

Eugene.

 

[A. Neumaier]

Try to model time and frequency of a classical signal by a joint probability distribution, and you'll see that it is impossible.

Quantum phenomena are generally like this.

 

[vanhees71]

I think the confusion comes from the fact that one has to rethink what the sensible definitions of probabilities are within quantum theory compared to what they are within classical physics, and there are several ways to talk about it in terms of mathematics. One way is to use classical probability theory as based on the axiomatic framework by Kolmogorov. Another way is to see quantum theory as a generalized form of probability theory (and even logic). From a physical point of view, I think both ways are finally the same thing, if you are willing to accept the minimal interpretation, which asserts that all QT is about is to describe the probabilities for the outcome of measurements, and that there's no other thing objectively knowable about nature than these probabilities.

The traditional way with usual probabilities a la Kolmogorov should be roughly that there are states, reprensented by self-adjoint positive semidefinite operators in a Hilbert space with trace 1, the statistical operator. Physically they represent possible "preparations of the system", e.g., setting up a proton (or rather a beam of protons) with a quite well defined momentum. Then there are observables that are also represented by self-adjoint operators on the Hilbert space. The possible outcomes are the eigenvalues of these operators and the probabilities can be calculated with help of the statistical operator.

It is not possible to determine the values of all possible observables at once but only of sets of compatible ones, which are those operators which have common complete sets of eigenvectors, and you can thus define only probabilities in a classical sense by choosing such sets of compatible observables. At the same time if you have a complete set of observables, for which the common eigenspaces are all onedimensional, you prepare the system such that of them have a determined value, then the system state is completely defined as the pure state given by this common eigenvector, i.e., the state is then given by the unique projection operator to this 1-d common eigenspace given by the corresponding eigenvalues.

There are extensions to this idealized "projection valued" probability measures to more realistic descriptions of weaker measurement descriptions, based on the approach defining probabilities by positive operator valued measures, but that's not necessary to start understanding quantum theory.

 

[Killtech]

However, how can we be sure that the same logical laws apply to measurements of the electron's position and momentum? These measurements are inherently probabilistic and can't be done simultaneously. So, it is only natural to assume that these measurements can be ruled by a different logic.

It is not possible to determine the values of all possible observables at once but only of sets of compatible ones [...]

Okay, let's stop at this premise here because i found that to be the biggest difference when i started thinking about how modelling of quantum experiments it classical PT would work. It takes
the very different perspective here in that the only observables it accepts are detector readings of the experiments and these are always well-defined and all of them can be extracted simultaneously at all times (same for quantum experiments as for any other). That's the huge problem i have in producing an contradiction with this approach. PT simply doges this core premise of QM interpretations entirely. And I'm not sure it's not needed to describe or predict any experiment per se.

The problem is that hypothetical quantities/observables like the position or momentum of a point particle electron require already a good portion of interpretation in order to even define what those are supposed to mean. On the other hand PT starts blank so those things don't have meaning until you can define them in a PT model describing a quantum experiment.

PeroK made a similar argument and my more detailed response to that is here:
https://www.physicsforums.com/threads/quantum-physics-vs-probability-theory.981195/#post-6268363 (4th post)

 

[Stephen Tashi]

the very different perspective here in that the only observables it accepts are detector readings of the experiments and these are always well-defined and all of them can be extracted simultaneously at all times (same for quantum experiments as for any other).

Presumably you don't mean to say all properties of a physical system can be measured simultaneously, so what you mean by "all of them"?

That's the huge problem i have in producing an contradiction with this approach.

What is a huge problem? What approach are you talking about? What do you desire to contradict?

PT simply doges this core premise of QM interpretations entirely.

If you're saying that probability theory doesn't deal with physics, that's obvious, isn't it?

And I'm not sure it's not needed to describe or predict any experiment per se.

What you mean by the double negative "not sure it's not needed"?

The problem is that hypothetical quantities/observables like the position or momentum of a point particle electron require already a good portion of interpretation in order to even define what those are supposed to mean. On the other hand PT starts blank so those things don't have meaning until you can define them in a PT model describing a quantum experiment.

As I interpret your basic question , it is "How do I define a probability space that can be used state the results of quantum mechanics?"

My thoughts:

1) Almost all real life problems require more than one probability space to model them. So an attempt to find a single probability space for all of QM is unrealistic.

2) Physicists rarely formulate theories as explicit probability models. They often discuss probability without defining a probability space. (Just look at expositions of statistical mechanics to see this style in action.) In addition to the difficulty of defining a probability space for some aspect of QM, you have the cultural problem that physicists don't want to do things this way. The focus in physics is on phenomena that cause (or can be used to calculate) probabilities. The probabilities pop-up and defining a probability space for them is an afterthought.

 

[Killtech]

Try to model time and frequency of a classical signal by a joint probability distribution, and you'll see that it is impossible.

of course you can model a classical signal in PT but how you can model it is not entirely free to chose. let's take a EM-wave signal. with corresponding detectors one can measure all of it Fourier series coefficients independently and thus extract all the information it holds. the information each Fourier coefficient hold is independent from all the other information thus it is irreducibile and must be modeled in the case the signal is allowed to interact with something that reacts differently to different wavelengths (e.g. water molecules) - otherwise eliminating such information will break the models predictive power. therefore to model it properly each Fourier coefficient will be modeled as a random variable with the complete signal being a joint probability distribution of all coefficients. of course the underlying physics is deterministic thus so will the stochastic process. however if the starting singal is not known, then the joint probability distribution won't be trivial but depend on what the possible forms the input signal is assumed to be. And of course such a model still remains boring from a PT point of view unless you let the unknown signal interact with something to allow the stochastics to really come into play.

therefore i'd argue it is very much possible.

 

[vanhees71]

I also don't see what precisely the problem is. You can only describe by probability theory what is clearly definable, and for that you need already a theory. QT tells us what we can describe probabilistically, and that's the outcome of measurements that are really feasible and not fictitious measurements that cannot be really made.

For massive particles it makes indeed sense to define observables like position, momentum, and polarization, various charges (electric charge, strangeness, baryon number, lepton number, etc.). The theory also tells us which variables can be precisely determined (again I stick to the most simple case of idealized precise measurements decribed by standard textbook quantum theory, the socalled projection measurements a la von Neumann). Given the state $\hat{\rho}$ a complete compatible set of observables $A_1,\ldots,A_n$, represented by a corresponding set of commuting self-adjoint operators $\hat{A}_1,\ldots,\hat{A}_n$ and common orthonormal eigenvectors $|a_1,\ldots,a_n \rangle$, a possible (in this case complete) measurement determining the values of these observables leads to the result $(a_1,\ldots,a_n)$ with a probaility (Born's Rule)
$$P(a_1,\ldots,a_n|\hat{\rho})=\langle a_1,\ldots a_n|\hat{\rho}|a_1,\ldots,a_2 \rangle.$$
That's a well-defined probability description for a (in principle) feasible measurement according to QT. Nowhere do I need extensions of the probability theory to describe the probabilities such measurement outcomes leading to a prediction about the statistics of the outcomes of such measurements.

As I said, there are extensions of the formalism to describe socalled weak measurements, where you also can determine the values of incompatible observables in terms of incomplete measurements, i.e., where you measure not all variables with arbitrary precision. This is often anyway the case, e.g., if a detector has only a finite efficiency in detecting a particle or a limited resolution in the particle's position or for its momentum etc. But first let's discuss the standard textbook QM, before we come to these more complicated (though often also more realistic) descriptions of measurements.

 

[A. Neumaier]

of course you can model a classical signal in PT but how you can model it is not entirely free to chose. let's take a EM-wave signal. with corresponding detectors one can measure all of it Fourier series coefficients independently and thus extract all the information it holds.

One can model the signal, but by Nyquist's theorem the information it holds doesn't include perfect information about duration and frequency.

therefore i'd argue it is very much possible.

Well, you can't measure or even define duration and frequency of a signal jointly to arbitrary accuracy. The product of the uncertainties is bounded from below just as in Heisenberg's uncertainty relation.

 

[meopemuk]

To convince myself that there are pairs of observables that cannot be measured simultaneously I use the following argument: Take $S_x$ and $S_y$ projections of particle's spin. In order to measure $S_x$ I have to have a Stern-Gerlach device oriented along the x-axis. In order to measure $S_y$ I have to have a Stern-Gerlach device oriented along the y-axis. However, it is not possible to have a device oriented along both axes simultaneously. So, there is no way to measure $S_x$ and $S_y$ at once.

Eugene.

 

[Killtech]

Presumably you don't mean to say all properties of a physical system can be measured simultaneously, so what you mean by "all of them"?

I also don't see what precisely the problem is. You can only describe by probability theory what is clearly definable, and for that you need already a theory. QT tells us what we can describe probabilistically, and that's the outcome of measurements that are really feasible and not fictitious measurements that cannot be really made.

Okay, sorry i used the term "observable" a little too loosely when i tried to make the connection with PT and it got mixed up, my bad.

Maybe i should rephrase my original question. probabilty theory defines what a random experiment is and because the outcomes of any performed quantum experiment are always well defined and their results are reproducible they fulfill this definitions criteria and as such become subject to PT treatment. Please, let's establish this as a starting point of my question first.

Furthermore PT doesn't have the exact concept of observables but describes everything in terms of outcomes - which makes them inherently experiment specific - i.e. they are only as general as your experimental setup allows them to be. while both terms aim to define what is measurable/observable they are technically quite different and i shouldn't have mixed them up. Now, outcomes for a particular experiment can't be defined analoge to observables. instead usually the simplest way to to identify each outcome is by a vector of everything observable - in the case of a physical experiment that would be the readings of all detectors and all experimental setup parameters (e.g. geometry) and what not.

with that said, let's consider the Stern-Gerlach experiment:

To convince myself that there are pairs of observables that cannot be measured simultaneously I use the following argument: Take $S_x$ and $S_y$ projections of particle's spin. In order to measure $S_x$ I have to have a Stern-Gerlach device oriented along the x-axis. In order to measure $S_y$ I have to have a Stern-Gerlach device oriented along the y-axis. However, it is not possible to have a device oriented along both axes simultaneously. So, there is no way to measure $S_x$ and $S_y$ at once.

I am well familiar with non-commuting observables. But i am trying to express the same thing via PT. Thus i would define the Stern Gerlach experiment sample space as all possible detector screen results cross product with all possible orientations of each SG-device and filter in the sequence - you need an experiment with at least 2 devices to be able to have the the non-commuting relation between $S_x$ and $S_y$ represented in PT because otherwise it is irrelevant for the outcome. That relation will then be embedded in the probability measure on that space. Therefore PT should always perfectly respect all commutation relations between observables - however as this is done intrinsically it becomes hard to express it in its formalism.

 

[vanhees71]

The point is that QT provides only probabilities for experiments you can really perform. In the case of the SGE it's indeed obvious that you can only measure (precisely in the sense of a von Neumann projection measurement) the component of the spin in one direction, given by the direction of the magnetic field in the SGE setup. Thus the only sensible random experiment is the measurement of one spin component.

It's also obvious that with the SGE measurement of this spin component you change the state of the particle. Say, you have prepared to particle to have a certain spin-z component $\sigma_z = +\hbar/2$, and now you measure the spin-x component. Then your particle is randomly deflected up or down (with 50% probability each) and depending on which direction the particle went it now has either well determined $\sigma_x=+\hbar/2$ or $\sigma_x=-\hbar/2$. So now you cannot determine the probability ot find either of the two possible $\sigma_y$-values (provided the particle is prepared such that $\sigma_z=+\hbar/2$, because now your particle has a determined $\sigma_x$ (and you know which value of the two possible values $\pm \hbar/2$ due to the direction to which it was reflected). So you cannot determine the outcome of two incompatible observables at once.

Of course, what you can do is you can always prepare an ensemble of particles with determined $\sigma_z=\hbar/2$ and then measure anyone spin-component you like, but with one experiment only one and not several components in non-parallel directions at once.

So QT describes precisely the probabilities for the outcomes of measurements of any observable you like (or any set of compatible observables you like).

 

[Stephen Tashi]

Thus i would define the Stern Gerlach experiment sample space as all possible detector screen results cross product with all possible orientations of each SG-device and filter in the sequence -

As I understand your idea, if we confine ourselves to two orientations $a,b$ and do two measurements, the outcomes in the probability space could be denoted:

($a$ is the orientation, result is Up, $a$ is the orientation, result is Up)
($a$ is the orientation, result is Up, $a$ is the orientation, result is Down)
($a$ is the orientation, result is Up, $b$ is the orientation, result is Up)
($a$ is the orientation, result is Up, $b$ is the orientation, result is Down)
($a$ is the orientation, result is Down, $a$ is the orientation, result is Up)
($a$ is the orientation, result is Down, $a$ is the orientation, result is Down)
($a$ is the orientation, result is Down, $b$ is the orientation, result is Up)
($a$ is the orientation, result is Down, $b$ is the orientation, result is Down)

($b$ is the orientation, result is Up, $a$ is the orientation, result is Up)
($b$ is the orientation, result is Up, $a$ is the orientation, result is Down)
($b$ is the orientation, result is Up, $b$ is the orientation, result is Up)
($b$ is the orientation, result is Up, $b$ is the orientation, result is Down)
($b$ is the orientation, result is Down, $a$ is the orientation, result is Up)
($b$ is the orientation, result is Down, $a$ is the orientation, result is Down)
($b$ is the orientation, result is Down, $b$ is the orientation, result is Up)
($b$ is the orientation, result is Down, $b$ is the orientation, result is Down)

If probabilities are assigned to the outcomes, we could compute the marginal probability of the event "The experimenter decides to use the orientation $b$ in the second measurement". We might choose to assign probabilities by assuming the experimenter picks an orientation at random by flipping a fair coin.

Therefore PT should always perfectly respect all commutation relations between observables - however as this is done intrinsically it becomes hard to express it in its formalism.

I don't see any problem with assigning probabilities to sequences of measurements and outcomes. Do you object to the fact that such a model portrays stochastic behavior by the experimenter?

 

[PeroK]

As I understand your idea, if we confine ourselves to two orientations $a,b$ and do two measurements, the outcomes in the probability space could be denoted:

($a$ is the orientation, result is Up, $a$ is the orientation, result is Up)
($a$ is the orientation, result is Up, $a$ is the orientation, result is Down)
($a$ is the orientation, result is Up, $b$ is the orientation, result is Up)
($a$ is the orientation, result is Up, $b$ is the orientation, result is Down)
($a$ is the orientation, result is Down, $a$ is the orientation, result is Up)
($a$ is the orientation, result is Down, $a$ is the orientation, result is Down)
($a$ is the orientation, result is Down, $b$ is the orientation, result is Up)
($a$ is the orientation, result is Down, $b$ is the orientation, result is Down)

($b$ is the orientation, result is Up, $a$ is the orientation, result is Up)
($b$ is the orientation, result is Up, $a$ is the orientation, result is Down)
($b$ is the orientation, result is Up, $b$ is the orientation, result is Up)
($b$ is the orientation, result is Up, $b$ is the orientation, result is Down)
($b$ is the orientation, result is Down, $a$ is the orientation, result is Up)
($b$ is the orientation, result is Down, $a$ is the orientation, result is Down)
($b$ is the orientation, result is Down, $b$ is the orientation, result is Up)
($b$ is the orientation, result is Down, $b$ is the orientation, result is Down)

If probabilities are assigned to the outcomes, we could compute the marginal probability of the event "The experimenter decides to use the orientation $b$ in the second measurement". We might choose to assign probabilities by assuming the experimenter picks an orientation at random by flipping a fair coin.
I don't see any problem with assigning probabilities to sequences of measurements and outcomes. Do you object to the fact that such a model portrays stochastic behavior by the experimenter?

I'm struggling to see what this has to do with QM. The essence of QM is as follows.

Imagine we have a particle at a source, $S$. It then may pass through either of two intermediate points - perhaps two slits in the double-slit experiment - let's call these $P1$ and $P2$. It then may or may not end up at point $X$ on a screen.

We can calculate the following:

$p(P1|S), p(P2|S)$ - the probability that a particle from the source passes through point
$P1, P2$ respectively.

We can also calculate:

$p(X|P1), p(X|P2)$ - the probability the particle impacts point $X$ on the screen, given it passed through points $P1, P2$.

Classically, we have:

$p(X|S) = p(X|P1)p(P1|S) + p(X|P2)p(P2|S)$

But, in nature this equation fails. Instead, the correct equation involves complex probability amplitudes:

$\psi(X|S) = \psi(X|P1)\psi(P1|S) + \psi(X|P2)\psi(P2|S)$

Where $p = |\psi|^2$. I.e. the probability is the modulus squared of the amplitude; and the amplitudes combine essentially as classical probabilities.

This is the heart of QM and the point where classical PT alone fails. If that's not what this thread is about, then I don't know what it is about.

 

[vanhees71]

The quantum description is a bit different. You have a particle of which you measure a spin component, and you know before the measurement that the particle is prepared in a state, where the spin-z component is determined to be $\sigma_z=\hbar/2$. The state is then a pure state and described by the projection operator
$$\hat{\rho}=|\sigma_z=\hbar/2 \rangle \langle \sigma_z=\hbar/2|,$$
where $\sigma_z=\hbar/2$ is an arbitrary normalized eigenvector of the spin-z operator $\hat{s}_z$ with eigenvalue $\hbar/2$.

Now suppose you measure with help of an SGE the spin-z component. Then the probability to find the possible values, $\pm \hbar/2$ are
$$P(\sigma_z=\pm \hbar|\hat{\rho})=\langle \sigma_x=\pm \hbar|\hat{\rho}|\sigma_x=\pm \hbar/2 \rangle = |\langle \sigma_x=\pm \hbar/2|\sigma_z=\hbar/2 \rangle|^2=1/2,$$
i.e., with 50% probality you get the result $\sigma_x=\hbar/2$ and with 50% probability $\sigma_x=-\hbar/2$.

 

[Killtech]

uh, too many people responding at once... and it's getting time intensive to answer fast enough ;)

[...] I don't see any problem with assigning probabilities to sequences of measurements and outcomes. Do you object to the fact that such a model portrays stochastic behavior by the experimenter?

Yeah, your description is quite along what i am trying to do. as for the stochastic behavior of the experimenter, hehe, i haven't though of it like that. i would normally only consider events where the experimental setup is known (therefore with an deterministic experimenter), but indeed the model would allows to also model more then that. this isn't actually a stupid thing to do because if you take only someones actual measurement results as known information and assume the experimental setup to be unknown it allows you to reverse engineer the setup as best as possible. it will also allow you to calculate the likelihood of whether measurement data was altered and in which way.

I'm struggling to see what this has to do with QM. The essence of QM is as follows.

Imagine we have a particle at a source, $S$. It then may pass through either of two intermediate points - perhaps two slits in the double-slit experiment - let's call these $P1$ and $P2$. It then may or may not end up at point $X$ on a screen.

From a PT point of view your are probably doing a mistake with the assumption of a particle behavior even before establishing the probably space of the experiment.

PT modeling works kind of in reverse to QM in that you begin with identifying all experimental outcomes first and then reverse engineer a model that is capable to produce these outcomes whereas QM starts with an established model and yields predictions for the experiment. in a sense in PT you do not presume to know what you are modelling/dealing with. this is an important aspect to keep in mind - and perhaps it could serve as a tool to challenge assumptions.

therefore with PT you don't assume to know that you are dealing with a particle - you have to derive it from the outcomes the experiment it produces. instead you are dealing only with the information directly observable (setup, detector results). but with what you have shown exposes an obvious fact: PT (probably) cannot model elementary particles just as... point like particle objects. in fact complexity of outcomes rather indicates that they have far more degrees of freedom available then just 3+3 for impuls and location. though this isn't really news as QM describes them via a wave function that has infinity many degrees of freedom for a reason. therefore if PT approach to QM modeling would work i would expect it to recover at least parts of the wave function from QM - the minimal information stored in a QM quantum state that is required to predict outcome probabilities correctly. However if it would work it would likely come into conflict with most interpretations of QM.

 

[PeroK]

From a PT point of view your are probably doing a mistake with the assumption of a particle behavior even before establishing the probably space of the experiment.

PT modeling works kind of in reverse to QM in that you begin with identifying all experimental outcomes first and then reverse engineer a model that is capable to produce these

In QM you must identify the probability amplitude space first; not the probability space. That is your mistake. And, since we are talking about physics that is backed by established experimental results, then your mistake is the critical one; and prevents you from understanding QM.

I'm quite happy to understand classical PT on the one hand and QM PAT (probability amplitude theory) on the other. If you restrict yourself to classical PT, then you can have no complaints if QM and the behaviour of the natural world eludes you.

whereas QM starts with an established model and yields predictions for the experiment.

This is called "physics".

 

[PeroK]

uh, too many people responding at once... and it's getting time intensive to answer fast enough ;)

Before you go any further, you should read this:

https://www.scottaaronson.com/democritus/lec9.html

All would be well, in fact, if some bright mathematician in the 19th century had invented complex probability spaces before QM forced it out. Then, QM would rely on classical PT. It's just that classical PT would have been already enriched by complex probability amplitudes!

Your objections seem to me similar to rejecting GR because it doesn't use Euclidean geometry.

Why should nature restrict itself to the classical subset of PT that humans worked out before QM showed how it could be generalised?

 

[Stephen Tashi]

Imagine we have a particle at a source, $S$. It then may pass through either of two intermediate points - perhaps two slits in the double-slit experiment - let's call these $P1$ and $P2$. It then may or may not end up at point $X$ on a screen.

We can calculate the following:

$p(P1|S), p(P2|S)$ - the probability that a particle from the source passes through point
$P1, P2$ respectively.

I don't see that this has anything to do with a limitation of probability theory. If indeed, the slit through which the particle passes is measured then you'd get classical results. The problem is with the physical model. The particle does not pass though only one slit or the other unless it is observed to do so. The fact that a given probability model does not agree with experimental data doesn't demonstrate a limitation of probability theory. It demonstrates a limitation of the concepts used in the model.

 

[PeroK]

I don't see that this has anything to do with a limitation of probability theory. If indeed, the slit through which the particle passes is measured then you'd get classical results. The problem is with the physical model. The particle does not pass though only one slit or the other unless it is observed to do so. The fact that a given probability model does not agree with experimental data doesn't demonstrate a limitation of probability theory. It demonstrates a limitation of the concepts used in the model.

It's absolutely a limitation in probability theory. Similar to the limitation in number theory if you confine yourself to the natural numbers. Or, the limitations in geometry if you confine yourself to Euclidean geometry.

Or, of course, the real numbers without the algebraic closure of the complex numbers. Which is probably the most relevant!

Try:

https://www.scottaaronson.com/democritus/lec9.html

 

[Stephen Tashi]

Why should nature restrict itself to the classical subset of PT that humans worked out before QM showed how it could be generalised?

I look at it this way. Probability Theory does not cover how to solve differential equations. Differential equations are useful for describing the phenomena of Nature. So, in that sense, the omission of differential equations is a limitation of probability theory.

Likewise, probability theory does not cover how to analyze physical situations using (in Griffith's words) "pre-probabilities" such as wave functions. So the omission of pre-probability techniques is also a limitation of probability theory.

Whether pre-proability methods are a "generalization" of probability theory is a subjective question. To me, this depends on how such techniques are presented. If the presentation leads in an organized way to generating probability spaces, then I'd call them a generalization. However, this is not how they are usually presented!

 

[PeroK]

I look at it this way. Probability Theory does not cover how to solve differential equations. Differential equations are useful for describing the phenomena of Nature. So, in that sense, the omission of differential equations is a limitation of probability theory.

You really can see no relationship here? Probability amplitudes and probabilities are as distantly related as probabilities and differential equation?

 

[Stephen Tashi]

You really can see no relationship here? Probability amplitudes and probabilities are as distantly related as probabilities and differential equation?

State the abstract framework for probability amplitudes and define the probability spaces they imply and maybe you can convince me otherwise. Do this without referring to applications of the theories to physics. Then we'll see how mathematically related the theory of probability amplitudes is to the theory of probabiliy.

If we were to consider applications instead of theory then differential equations, probability amplitudes, and probability theory can be used together. From the point of view of certain applications, differential equations and probability theory are closely related.

 

[PeterDonis]

State the abstract framework for probability amplitudes and define the probability spaces they imply and maybe you can convince me otherwise...Then we'll see how mathematically related the theory of probability amplitudes is to the theory of probabiliy.

So "probabilities are the squared moduli of probability amplitudes" doesn't count?

 

[Stephen Tashi]

So "probabilities are the squared moduli of probability amplitudes" doesn't count?

A probability isn't completely defined until it's probability space is specified. So "probabilities are the squared moduli of probability amplitudes" doesn't define a context for using probability amplitudes unless we assume we begin with a probability space and define probability amplitudes in terms of probabilities on that space. That would make probability amplitudes a function of probabilities, not a generalization of probabilities.

Isn't the usual approach to probability amplitudes to define them in the context of a mathematical structure more complicated that simply a set of outcomes and its sigma algebra?

 

[atyy]

For finite dimensional systems, Holevo, Statistical Structure of Quantum Theory, p1-2 states:

Classical
1. the set of events $A \subset \Omega$ forms a Boolean algebra,
2. the set of probability distributions $P = [p_{1}, . . ., p_{N}]$ on $\Omega$ is a simplex, that is a convex set in which each point is uniquely expressible as a mixture (convex linear combination) of extreme points,
3. the set of random variables $X = [\lambda_{1} ... \lambda_{N}]$ on $\Omega$ forms a commutative algebra (under pointwise multiplication).

Quantum
1. the quantum logic of events is not a Boolean algebra; we do not have the distributivity law $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$ does not hold. Consequently there are no "elementary events" into which an arbitrary quantum event could be decomposed;
2. the convex set of states is not a simplex, that is, the representation of a density matrix as a mixture of extreme points is non-unique;
3. the complex linear hull of the set of observables is a non-commutative (associative) algebra.

 

[Stephen Tashi]

For finite dimensional systems, Holevo, Statistical Structure of Quantum Theory, p1-2 states:

I understand the books statements as statements about properties of some mathematical structures and their applications, but the quoted text lacks definitions for these structures.

Classical
1. the set of events $A \subset \Omega$ forms a Boolean algebra,

In probabiity theory, the usual definition of an "event" is that it is a set in a sigma algebra of sets. To define an event as something that will form a boolean algebra, we need an event to be a proposition. So apparently the event $A$ as a set is identified with the propositional "The outcome of the experiment is a member of $A$". So we are dealing with a particular application of probability theory.

2. the set of probability distributions $P = [p_{1}, . . ., p_{N}]$ on $\Omega$ is a simplex, that is a convex set in which each point is uniquely expressible as a mixture (convex linear combination) of extreme points,

Is $\Omega$ the set of outcomes $\Omega$ used in the definition of a probability space as $(\Omega, \mathscr{F}, \mu)$? Does the notation for $P$ indicate that the set of outcomes is finite? Probability distributions are special cases of the more general concept of a probability measure mentioned in the definition of a probability space. A finite set of outcomes is also a special case.

3. the set of random variables $X = [\lambda_{1} ... \lambda_{N}]$ on $\Omega$ forms a commutative algebra (under pointwise multiplication).

That would be a theorem of probability theory that follows from the definition of a random variable as a function whose domain is the set $\Omega$ of outcomes.

Quantum
1. the quantum logic of events is not a Boolean algebra; we do not have the distributivity law $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$ does not hold. Consequently there are no "elementary events" into which an arbitrary quantum event could be decomposed;

We have the problem of defining an "event". Apparently it is no longer a proposition. So what is it? My guess: We are discussing a non-distributive lattice. An "event" is an element of the lattice. Instead of "and" and "or", we are using the operations of "meet" and "join".

2. the convex set of states is not a simplex, that is, the representation of a density matrix as a mixture of extreme points is non-unique;

What's the definition of "the set of states"? As to density matrix, what's the definition of "density"? If 2. under the Quantum case is analogous to 2 under the Classical case then a "state" should be analgous to a probability distribution. So one guess is that a "state" is a complex valued function whose domain is the set of elements of the non-distributive lattice.

However, according the Wikipedia article on "Quantum Probability", https://en.wikipedia.org/wiki/Quantum_probability , a state is a linear functional who domain is the set of elements in a star-algebra and whose range is the set of complex numbers.

3. the complex linear hull of the set of observables is a non-commutative (associative) algebra.

We're missing the mathematical definition of observables and what operations on them make them an algebra. What's the connection between an "observable" and an "event"?

 

[vanhees71]

I'm really puzzled what's still the problem. The probabilities in QT are the usual probabilities of Kolmogorov's axioms if you restrict yourself to von Neumann measurements, i.e., the measurement of a set of compatible observables. The probability space is the possible outcome of measurements of these observables, and the probabilities are given by Born's rule. That's the meaning of the quantum state.

If you insist on going beyond this, i.e., asking for a framework considering all possible measurements, i.e., of all observables, then you need to extend the logic and the probability theory beyond the standard features, as explained by @atyy in #33 .