I Quantum physics vs Probability theory

[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 .

 

[A. Neumaier]

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

Sets form a Boolean algebra under union and intersection.

We have the problem of defining an "event". Apparently it is no longer a proposition.

In quantum logic, an event is an orthogonal projector. These form a modular lattice, not a Boolean one.

What's the definition of "the set of states"? As to density matrix, what's the definition of "density"?

The set of states is the set of positive linear functionals. In finite dimensions, they are in 1-1 correspondence with density matrices. This is just a label; no definition of density is needed.

We're missing the mathematical definition of observables and what operations on them make them an algebra.

An observable is a positive operator-valued measure.

 

[vanhees71]

Usually observables are defined as being represented by self-adjoint operators and the possible values the eigenvalues of these operators. This refers to ideal von Neumann measurments.

POVMs are a (quite recent) addition to describe more complicated "weak measurements", being often more realistic though an unnecessary complication for discussing fundamental questions like the one discussed here.

Before you haven't understood the simpler case of precise measurements, it doesn't make sense to try to understand more complicated subjects.

That's an analogous further step as the one from the idealized definition of observables in classical Hamiltonian mechanics compared to the more realistic description of it in terms of phase-space distribution functions in classical statistical mechanics.

 

[Stephen Tashi]

Sets form a Boolean algebra under union and intersection.

It's interesting that a boolean algebra "of logic" ( https://en.wikipedia.org/wiki/Boolean_algebra) is defined differently than a boolean algebra "of sets".

In quantum logic, an event is an orthogonal projector. These form a modular lattice, not a Boolean one.

Ok, but my line of inquiry concerns whether terms like "quantum logic" and "quantum probability" have mathematical definitions - or are they merely informal terms that describe qualitative properties of how QM calculations are peformed?

The statement that an event is an orthogonal projector gives a property of the thing called an "event", but it doesn't define an event mathematically until there is some context is defined for a projection of something on something else. We should begin with a Hilbert space, correct?

 

[Stephen Tashi]

I'm really puzzled what's still the problem.

If that refers to my questions, the problem is to show that "quantum logic" or "quantum probability" or "probability amplitudes" are organized mathematical topics that generalize ordinary probability theory. The alternative to that possibility is that these these terms are not defined in some unified mathematical context, but are informal descriptions of aspects of calculations in QM.

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 .

Post #33 doesn't describe a unified mathematical system. The unified mathematical system that begins with a Hilbert space and defines states and observables in that context is apparently the background for post #33. Within that context, what is the definition of "quantum logic"? What is the definition of a "quantum probability"? Is there a mathematical definition of the Kolmogorov probability spaces associated with the Hilbert space? - or are the associated probabiity spaces a matter of interpretation and application of the theory?

 

[PeterDonis]

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.

No, this is not correct. If the space of probability amplitudes is defined, and we know that probabilities are squared moduli of amplitudes, then the space of probabilities is defined. You don't have to assume the probability space first.

 

[A. Neumaier]

POVMs are a (quite recent) addition to describe more complicated "weak measurements",

This is quite misleading.

POVM measurements were introduced in 1970, which was 45 years after Heisenberg's 1925 paper initiating modern quantum physics. A very readable account was given 4 years later in

• S.T. Ali and G.G. Emch, Fuzzy observables in quantum mechanics, J. Math. Phys. 15 (1974), 176--182.

Since this paper appeared, another 45 years passed. Thus it is not appropriate to call them ''a (quite recent) addition''.

Moreover, they are not used to describe weak measurements (which is a special class of continuous measurements on single quantum objects). They are needed to describe quite ordinary experiments (such as Stern-Gerlach or the double slit) without making the traditional textbook idealizations. See the books mentioned in another thread. They are indispensable in quantum information theory. Indeed, the well-known textbook

• M.A. Nielsen and I.L, Chuang, Quantum computation and quantum information, Cambridge Univ. Press, Cambridge 2001.

introduces them even before defining the traditional projective measurements that you advocate one should restrict attention to!

 

[A. Neumaier]

whether terms like "quantum logic" and "quantum probability" have mathematical definitions

The mathematical definition is that quantum logic is the logic specified by the algebraic properties of orthomodular lattices. Maybe you'd read the book 'Quantum Logic' by Svozil before ranting about the subject. Similarly, you'd first read the Wikipedia article on quantum probability, say, before making your unfounded claims.

 

[Stephen Tashi]

No, this is not correct. If the space of probability amplitudes is defined, ...

Yes, if it is defined. What is the definition of "the space of probability amplitudes"?

 

[cosmik debris]

The mathematical definition is that quantum logic is the logic specified by the algebraic properties of orthomodular lattices. Maybe you'd read the book 'Quantum Logic' by Svozil before ranting about the subject. Similarly, you'd first read the Wikipedia article on quantum probability, say, before making your unfounded claims.

I think characterising Stephen's posts as ranting (lengthy, angry and impassioned) is inappropriate.

Cheers

 

[PeterDonis]

What is the definition of "the space of probability amplitudes"?

The space of complex components in the chosen basis of all possible state vectors. The basis is chosen based on the observable you want to measure.

 

[A. Neumaier]

I think characterising Stephen's posts as ranting (lengthy, angry and impassioned) is inappropriate.

Lengthy uninformed criticism looks like a rant. Anger or passion are not necessarily implied in this word:

A diatribe, also known less formally as rant, is a lengthy oration, though often reduced to writing, made in criticism of someone or something, often employing humor, sarcasm, and appeals to emotion.

(from https://en.wikipedia.org/wiki/Diatribe)

Stephen Tashi discounted the introductory informal short description, quoted from a book that later explains everything in detail, as

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

and then gave long superficial reasons (of ignorance) for his assessment. He should have spent the time for writing his critique on reading the subsequent pages of the book.

 

[cosmik debris]

Lengthy uninformed criticism looks like a rant. Anger or passion are not necessarily implied in this word:

(from https://en.wikipedia.org/wiki/Diatribe)

Stephen Tashi discounted the introductory informal short description, quoted from a book that later explains everything in detail, as

and then gave long superficial reasons (of ignorance) for his assessment. He should have spent the time for writing his critique on reading the subsequent pages of the book.

OK, I understand that our definitions of rant maybe different, in my dictionary anger and passion are definite ingredients, additionally in my country rant is a pejorative term.

If everyone else thinks this is OK then I guess it is OK.

Cheers

 

[PeterDonis]

I understand that our definitions of rant maybe different

Yes, and @A. Neumaier explained how he was using the word, so any ambiguity that might have been present before has been removed.

Please focus on the substance of the points being debated instead of on people's choice of words.

 

[vanhees71]

If that refers to my questions, the problem is to show that "quantum logic" or "quantum probability" or "probability amplitudes" are organized mathematical topics that generalize ordinary probability theory. The alternative to that possibility is that these these terms are not defined in some unified mathematical context, but are informal descriptions of aspects of calculations in QM.
Post #33 doesn't describe a unified mathematical system. The unified mathematical system that begins with a Hilbert space and defines states and observables in that context is apparently the background for post #33. Within that context, what is the definition of "quantum logic"? What is the definition of a "quantum probability"? Is there a mathematical definition of the Kolmogorov probability spaces associated with the Hilbert space? - or are the associated probabiity spaces a matter of interpretation and application of the theory?

Well, to be honest, for me as a phenomenologically oriented theoretical physicist there was nowhere any need for other mathematical logic and probabilities than the usual ones. As I said repeatedly, as long as you look at what's really measured in the lab, there's no problem whatsoever with the well-defined standard mathematics. Rather than quantum logic, what's really an important extension to standard textbook quantum theory seems to be the extension of the idea of (idealized) von Neumann meausrements to the more comprehensive description by positive operator valued measurements, and as far as I understand this is also a well-defined mathematical subject.

 

[vanhees71]

This is quite misleading.

POVM measurements were introduced in 1970, which was 45 years after Heisenberg's 1925 paper initiating modern quantum physics. A very readable account was given 4 years later in

• S.T. Ali and G.G. Emch, Fuzzy observables in quantum mechanics, J. Math. Phys. 15 (1974), 176--182.

Since this paper appeared, another 45 years passed. Thus it is not appropriate to call them ''a (quite recent) addition''.

Moreover, they are not used to describe weak measurements (which is a special class of continuous measurements on single quantum objects). They are needed to describe quite ordinary experiments (such as Stern-Gerlach or the double slit) without making the traditional textbook idealizations. See the books mentioned in another thread. They are indispensable in quantum information theory. Indeed, the well-known textbook

• M.A. Nielsen and I.L, Chuang, Quantum computation and quantum information, Cambridge Univ. Press, Cambridge 2001.

introduces them even before defining the traditional projective measurements that you advocate one should restrict attention to!

Well, in my field we haven't ever needed POVMs at all. Where do you need them to understand the double-slit or Stern-Gerlach experiments?

Again, I'm not against the POVM formalism, but it's overcomplicating things if you start on a level where even the simpler and straight-forward case has been understood. Maybe it's a difference in the scientific culture in math vs. physics. The former tends to be taught in a deductive manner starting from the most general case and then treat the simple things as special cases (though in practice you cannot do this, because you have to start with the simple cases first, and I don't think that you can understand math in a purely Bourbakian approach) while the latter is an inductive empirical science.

 

[A. Neumaier]

Well, in my field we haven't ever needed POVMs at all.

Because there one doesn't care about the foundations. One would need them if one were to give a statistical interpretation to all the q-expectations of nonhermitian field products occurring in the derivation of quantum kinetic equations. They cannot be interpreted statistically in the pre-1970 measurement formalism.

Where do you need them to understand the double-slit or Stern-Gerlach experiments?

The original Stern-Gerlach experiment did (in contrast to its textbook caricature) not produce two well-separated spots on the screen but two overlapping lips of silver.
This outcome cannot be described in terms of a projective measurement but needs POVMs.

Similarly, joint measurements of position and momentum, which are ubiqiotpus in engineering practice, cannot be described in terms of a projective measurement.
Born's rule in the pre-1970 form does not even have idealized terms for these.

For the double slit without the common idealizations, which also needs a POVM treatment, see the book mentioned in the POVM thread.

Again, I'm not against the POVM formalism, but it's overcomplicating things if you start on a level where even the simpler and straight-forward case has been understood.

To motivate and understand Born's rule for POVMs is much easier (one just needs simple linear algebra) than to motivate and understand Born's rule in its original form, where all the fancy stuff about wave functions, probability amplitudes and spectral representations must be swallowed by the beginner.

Thus it is overcomplicating things if you start with probability amplitudes and spectral resolutions!

 

[Elias1960]

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 .

No, there is no need to do this. All you have to do is to care about accurate formulations of the propositions you make about some quantum systems.

Of course, if you, say, define the negation of "measuring X gives always result x" as "measuring X gives never result x", this operator "not" does not follow classical logic, thus, it defines some "quantum logic". This is essentially all that has to be said about such "generalizations" of classical logic: Care about what you say, and follow the rules of classical logic, and you will not need any quantum logic in quantum theory too.

Same for probability theory. You can use the space of elementary events proposed by Kochen and Specker (in their paper about the impossibility of hidden variables, where they have given that construction but rejected it as not giving what is usually assumed to be meant with "hidden variables"). This and caring about not violating classical logic in the own reasoning is sufficient to live with classical probability theory there too.

 

[Killtech]

Same for probability theory. You can use the space of elementary events proposed by Kochen and Specker (in their paper about the impossibility of hidden variables, where they have given that construction but rejected it as not giving what is usually assumed to be meant with "hidden variables"). This and caring about not violating classical logic in the own reasoning is sufficient to live with classical probability theory there too.

Thank you! I must admit I didn't know any detail about Kochen-Specker other then being a No-Go theorem before but their article answers my original question quite well. It was a very interesting read and in particular the formulation of the framework they used to approach the problem.

That said I found simple attempts to model any quantum experiment to become highly contextual. Though i think that in attempting to model a much more general setup or set of experiments this could be remedied.

But what i can't see any PT model to do is to allow to fully describe all state in terms observable results one way or another (assuming any such model to correctly describe the experiment it models i.e. yields correct predictions). So this kind of approach should not come into conflict with this theorem.

 

[Elias1960]

That construction on p.63 of the Kochen Specker paper was thought only as an illustration why one needs a more serious restriction for an adequate definition of "hidden variables" than a formula of some space $\Lambda$ which gives a quite trivial Kolmogorovian probability space for quantum theory too. It has been essentially ignored, so it is known only by those who have read the paper.

The QM pure states are defined in terms of observables, you observe a preparation measurement, then the observable result defines the eigenstate of the measured operator.

Instead, the interpretations are not restricted to describe the state in observable only terms. That would be an unreasonable restriction, motivated by nothing but positivism.

On the other hand, there is the objective Bayesian interpretation of probability (following Jaynes it is the "logic of plausible reasoning"). In some sense, it is not about anything hidden - it is about probabilities of things that make sense to us. But these things may be wrong (say a particular hypothesis about what has happened), maybe unrelated to anything real (like statements about what is true if a particular theory is true). So, this is not only about observables, and should not be, because we want to reason about how probable such things are and want to do this in a consistent way, and the rules of probability theory are what is appropriate for this.

 

[atyy]

On the other hand, there is the objective Bayesian interpretation of probability (following Jaynes it is the "logic of plausible reasoning"). In some sense, it is not about anything hidden - it is about probabilities of things that make sense to us. But these things may be wrong (say a particular hypothesis about what has happened), maybe unrelated to anything real (like statements about what is true if a particular theory is true). So, this is not only about observables, and should not be, because we want to reason about how probable such things are and want to do this in a consistent way, and the rules of probability theory are what is appropriate for this.

Just to point out that this is controversial. Even among Bayesians, many object to the objective Bayesian view. There are alternatives such as the subjective Bayesian view of de Finetti.

 

[Elias1960]

Just to point out that this is controversial. Even among Bayesians, many object to the objective Bayesian view. There are alternatives such as the subjective Bayesian view of de Finetti.

They both have their applications. If one wants to speculate beyond the information one objectively has, one may also want to do this in a logically consistent way. In this case, the subjective Bayesian point of view can be used.

The difference is roughly this: If you have no information about a dice which makes a difference between the numbers, then the objective Bayesian interpretation prescribes that one has to use $\frac{1}{6}$ for all of them. The subjective Bayesian interpretation makes no such prescriptions. You are free to speculate that 5 may be favored based on your subjective feeling. Whatever - there is not really a contradiction between them.

So I think the difference between objective and subjective Bayesians is in this question irrelevant. For physics, the objective view is clearly preferable. Already because it gives, for free, a base for the null hypothesis: If we have no information which suggests any causal connection between A and B we have to assume P(AB)=P(A)P(B). But also all that is named entropic inference (say, for the Bayesian variant of thermodynamics) depends on what can be said about the case when we have no information. In this case, objective Bayesians prescribe the probability distribution with the largest entropy, while subjective Bayesians tell us nothing at all.

 

[Killtech]

They both have their applications. If one wants to speculate beyond the information one objectively has, one may also want to do this in a logically consistent way. In this case, the subjective Bayesian point of view can be used.

well, as for original question i was actually looking for merely modeling the information we objectively have and wanted to know PT was in general always suitable for that. if it didn't that would be a very interesting thing to understand - but never mind that now.

My problem with QM is that for me all its interpretations add more confusion then they help to understand what we are actually modelling. having a solid framework to view problems from that is free of all the confusing assumptions of QM interpretations (the actual math isn't the confusing part) might be a good way to reflect and understand which particular aspects are causing the trouble.

for example QM interpretations stick to the idea of point like particles even though the entire math framework does everything it can to model as far as possible from that intuition. and never mind the idea of a point like charged particle is already incomprehensible and paradox on a classical level. It feels like QM get's this to work only because it's math framework cheats its interpretations and secretly gives up all such assumptions.

I hoped to maybe use PT to get this sorted out in my head; to understand what kind of information is there to be modeled - on the very abstract level of minimalist PT approach since it is very lightweight on axioms which makes it extremely general. In that context it would like to stay minimalist and not add any assumptions i don't absolutely need to get correct predictions. that makes me go with the most basic interpretations of probability.

 

[Killtech]

Having said that, I agree that a B level thread might not be the correct avenue for raising such an issue.

I am terribly sorry to have misunderstood this classification. is there a way I can remedy this mistake?

 

[Auto-Didact]

I am terribly sorry to have misunderstood this classification. is there a way I can remedy this mistake?

Ask a moderator to change it to A

 

[PeroK]

I am terribly sorry to have misunderstood this classification. is there a way I can remedy this mistake?

The moderators would have done so if they thought it was important enough.

The more important issue is that you don't understand QM:

My problem with QM is that for me all its interpretations add more confusion then they help to understand

for example QM interpretations stick to the idea of point like particles even though the entire math framework does everything it can to model as far as possible from that intuition.

This second quotation is, quite simply, nonsense. It does not reflect a failure of 100 years of QM development by the leading physicists of the 20th century. It reflects your failure, hitherto, to understand what QM is saying.

My concern is that we've indulged you in a fairly pointless exercise in analysing the foundations of QM vis-a-vis classical PT. Whereas, all along your issue is simply that of someone trying to learn QM for the first time and being confused by it.