# Quantum physics vs Probability theory

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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
Homework Helper
Gold Member
2018 Award
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
Homework Helper
Gold Member
2018 Award
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
Mentor
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
Gold Member
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
Gold Member
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
Mentor
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",

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!

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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"?

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
Mentor
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:
Wikipedia said:
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.

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
Mentor
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
Gold Member
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
Gold Member

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.