# Classical and Quantum probabilities

1. Mar 23, 2005

### seratend

Can we say that the classical (Kolgomorov axiomatization) and quantum probabilities results are different, when we consider only the statistical results (i.e. the probability of events)?
Can we really distinguish a classical statistical result from a quantum one (i.e. the born rules)?

What are the real differences between these 2 mathematical tools when we consider only the experimental results (the statistics or if you prefer the frequencies of the events)?

Seratend.

2. Mar 24, 2005

### bw

I don't know if I understand your question correctly, but I'll try.

According to usual probability rules, if A and B are two mutually exclusive events, then P(AUB) = P(A)+P(B). In words, if an event can occur via two mutually exclusive ways, then the probablity that the event occurs is the sum of the probability of each alternative.

Now consider the two slit experiment in quantum mechanics.

Let's say we see an electron on a certain spot on the screen. A is the event that the electron has arrived through the top hole, B is the event that it has passed through the bottom hole. A and B are two mutally exclusive alternatives. The meanings of P(A) and P(B) are self evident.

So as above P(AUB) =P(A)+P(B). AUB is the event that the electron has passed through exactly one hole.

Mathematically this is perfectly sound.

But in order to identify the left hand side of the equation as the probability of observing an electron we made a crucial PHYSICAL assumption informed by classical mechanics, which turns out to be wrong.

The assumption is that the electron must have gone through a hole and one hole only in order to arrive at the screen. That is, we mistakenly assumed the detection of the electron is equivalent to the event AUB, which is not!

The probability P(AUB) is STILL CORRECTLY given by the sum rule.Only that AUB does not represent the physical situation. P(AUB) = P(A)+P(B) still holds. But P(AUB) is NOT the probability of detecting the electron because the event that you detect an electron's is not given by AUB!

On the other hand, if you put detectors around the holes to track the electrons, then, indeed, the arrival of an electron at the screen implies it must have gone through exactly one hole. In this case the event AUB and the detection of the electron are equivalent.Hence the probability of observing an electron is P(A)+P(B)

The often made but incorrect claim that quantum mechanics invalidates the logical rule of "exclusion of the middle" basically stems from the same fallacy.
The rule always holds. Quantum mechanics merely says that the "either -or" assumption may not apply.

IMO there is no such thing as "quantum statistics" different from "classical" statistics. There is only one statistics and it applies even in quantum mechanics (OK, two if you consider Baysian v.s frequentist statistics). You only need to be careful with your physical assumptions when you use your
mathematical tools.

Last edited: Mar 24, 2005
3. Mar 25, 2005

### kleinwolf

Another difference between both is expressed by Bell's inequalities.

4. Mar 25, 2005

### SpaceTiger

Staff Emeritus
I would say that the difference, as we think we understand it, is that in classical systems the probabilities arise because we lack information. For example, in thermodynamics, we don't know the properties of each individual particle. If we did, we wouldn't need to talk about probability. In the popular interpretation of QM, however, the probabilities are a fundamental characteristic of nature and do not arise from a lack of information.

5. Mar 25, 2005

### seratend

I just want to see if we are able to distinguish the quantum and classical probability results (I would like to have a good feedback from this community).
I just separate the problem of the time evolution or simply the evolution of a probabilistic state from the probability results/statistics (i.e. the number of times we get the same event in the frequentist view).

By classical probability, I just mean the mathematical tool. I am not assuming that the system under study evolves under classical or quantum mechanical laws (i.e “the time evolution of the probability lax).

In order to avoid extra complications/confusions (as long as it is not needed), we may try to focus on the frequentist approach in this thread (used in both classical and quantum experiments to compute the probabilities).

It is a good example to start understanding the initial questions. You have already made some implicit assumptions, where I think you may mix the time evolution (the path of electron) with the results of the experiment: the spot of the electron on the screen.
In classical probability we have formally the event SPOT, the result of the experiment (the electron has been detected on the screen at a given position labelled by the event SPOT). We thus have the probability P(SPOT). If we introduce the classical random variable X_screen (the position of the electron on the screen, we have P(SPOT)=P(X_screen=SPOT).
By saying that, I am not saying that X_screen is the path of the electron, just the detected impact of the electron on the screen.
Therefore P(X_screen=spot_x) is the interference probability result of the experiment expressed in a classical probability form (where spot_x are the different possible locations on the screen).

Now, you have the legitimate right to ask for the conditional probability P(A) and P(B). But why do you want that P(AUB)=P(A)+ P(B)? Classical probabilities deal only with the results of experiments and not with the [classical/quantum] path of particle (implicit assumption to get P(AUB)=P(A)+P(B)).
Moreover, Classical probability does not forbid that, if you measure the “which slit the electron passes” event, the probability law is changed as in QM so that the interferences disappear. (We have an evolution of the probability law due to an active measurement at the slits as in QM: it is the freedom of classical probability as well as QM probability).

If I connect this classical approach to the QM approach, I may also, formally, compute the statistics at the slits *without* doing any measurement on the slits. I will therefore have:
* P(A)= |<slitA|psi>|^2
* P(A)=|<slitB|psi>|^2
* P(X_screen=SPOT)=|<SPOT|psi>|^2 where |SPOT> is the position of the spot on the screen
* P(X_screen=spot_x)= |<spot_x|psi>|^2 is the interference pattern (where spot_x is the position on the screen).

In both cases case if we do an active measurement on the slits, we loose the interference pattern on the screen (evolution natively included in the Qm formalism but external to the classical probability formalism).

Why do you want to use classical mechanics with classical probabilities. You are free to allow other evolutions (e.g. bohmian mechanics or whatever else): we are trying to see if we can differentiate the results of the laws.
I use the word classical probability, just to differentiate the way we compute the probabilities results in the QM or in a Kolgomorov axiomatic basis. If you prefer, we can use the word “Kolgomorov probability” instead of the word “classical probability” in order not to confuse the evolution of a system with the probability results.

May you develop? It is the centre of my question. Can we really distinguish quantum and classical statistics.
Moreover, Classical probability doe not intrinsically include the evolution of the probability law, while the QM formalism includes it natively. Because in classical probability the evolution of the law is external, can we say that it allows in fact a wider set of possibilities compared with the QM formalism and its forced law evolution (the measurement, not the SE time evolution)?

Seratend

6. Mar 25, 2005

### seratend

I think your are confusing the locality problem and the statistics by themselves. Bell's inequalities may be reproduced by the classical probability formalism:
E(a,b)= Int dPab(w)Aa(w)Bb(w) instead of the formula E(a,b)= Int dP(w)Aa(w)Bb(w).

Again: do not confuse "classical probabilty" with a classical system following the newtonian time evolution on an unkonw state. (I must admit that my terminology is not the best one, sorry :). As I already said in a previous post, we can rename classical probability into Kolgomorovian probability to avoid confusions.

Seratend.

7. Mar 25, 2005

### seratend

However, we must not confuse the lack of information with the evolution of the knowledge of the system. Once again, by classical probability, I just mean Kolgomorov probability. The increase/decrease of information (the evolution of the knowledge) of the system may be controlled by SE or a Newton equation.

In thermodynamics, we just can say that the observed statistics of some systems may be described by a newtonian type time evolution equation. However, for thoses systems, I would like to say that we cannot distinguish the obtained statistics (E, P,T, etc ..) from a quantum time evolution (i.e therefore my questions).

Seratend

8. Mar 25, 2005

### SpaceTiger

Staff Emeritus
Yes, but with Newton's equations, there is no limit to amount of information you can potentially obtain. There is in QM. This is important because:

In traditional Kolmogorov probabilities, we can make measurements of any of the set of multiple quantities at one time, but in QM that is not always possible. The statement, "What is the probability that this particle will have position x and velocity v?" is meaningful with classical probabilities, but not in QM because you can't measure both simultaneously.

Last edited: Mar 25, 2005
9. Mar 25, 2005

### seratend

You are right but you must take into account that QM includes the law evolution in the measurements (the "projection postulate") while the Kolgomorov probabilities do not (external to the axiomatic formulation). Therefore, in QM, because of the projection postulate (evolution of the law), you have 2 kind of observables, the ones that does not change the law upon a measure, and the others that change the law (e.g. your x and v).
However your are free to define equivalent random variables under the kolgomorov formalism if you add an add hoc probability law under a measurement: this is almost what is done with the bohmian mechanics formal formulation (i.e when you remove the interpretation phylosophical aspects ; ).

If you look more precisely at your statement “you can't measure both simultaneously”, you must see that it is only significant, if you have a law evolution of the statistics (“the projection postulate”). Therefore, If I add an external law evolution to the Kolgomorov probabilities (formally, I have the right to do that) that behaves as the projection postulate of QM, do I retrieve the same statistics?
i.e. “(Kolgomorov probability + add hoc projection law evolution) ~ (QM probabilities)?”

I think so, but I am not sure (existence problem). It is why I appreciate your and the others feedback in PF.

Seratend.

10. Mar 25, 2005

### seratend

I must add some precisions to my previous post. Instead of law evolution, it is more correct to say the evolution of the Kolgomorov probability space: after a measurement, we may formally consider the probability space updates into a new one through an ad hoc equivalent projection postulate (freedom of the Kolgomorov axiomatisation) to match the QM probability statistic updates (state evolution).
Note that we are also free (formally) to update the random variables instead of the probability space to match the state evolution of QM (analogue to the schroedinger and Heisenberg QM state representation).

Seratend.

11. Mar 25, 2005

### bw

This is exactly my point. I am not sure why you're bringing in evolutionary law in the first place.You can ask whether a certain mathematical/statistical model is appropiate in solving a physical problem. But that is the problem of the physicists, not that of the statistician or the mathematician. Operator theory is "correct" regardless whether the rules of quantim mechanics are or not.
I don't see the relevance of this comment. Of course I was talking about the EXPERIMENTAL OUTCOME (the classical event) of "detecting an electron". Probability can only be referred to the OUTCOMES of measurement. Not to the evolution(dynamics), which is completely deterministic(so I am not quite sure what do you mean by "the evolution of probability law")

I wasn't mentioning evolution(though I did later, implicitly in pointing out the inappropiateness of the sum rule) because it was irrelevent at that stage.
But I explicitly said that P(AUB)= P(A)+ P(B) gives the correct answer in our example ONLY IN SO FAR AS classical assumptions apply, which allows us to identify the event "electron detected" with AUB. The fact that it is NOT a justifiable assumption(which I also explicitly stated) is a physics problem. This was exactly the point I was making.

The equation P(AUB)=P(A)+ P(B)
is always correct. You just can't use it in this instance. So this is not a question of probability theory not giving you the right answer. It is because you use it inappropiately.

"Classical" probability does not tell you how to assign probabilities to elementary events. It only tells you how to compute the probabilities of certain types of composite events(those that can be expressed through Boolean operations) when the probabilities of elementary events are given.

The questions of what should be considered elementary events and how to assign probability to such events are specific to the field you study. In QM measuring a system in pure state (described by a wave function) is an elementary event (hence in the two slit experiment you can't just add the probabilities, the event that you detect an electron is elementary). Born's rule tell you how to assign probabilities when mesauring such a system.

Economists, sociologists and gamblers have other definitions for "elementary events" and other rules to assign probabilities for these events.
Whether these rules are justified or not is not the problem of statistics.

You use the same probability theory independent of what theory of physics you assume. Only that you have to be careful not to use the wrong formulae. Perhaps I wasn't making my pont clear enough. I was saying the sum rule is ALWAYS correct. In classical physics (if you track the electron) you get the correct answer because in this case "detecting an electron" is the same as the event AUB. But in QM you get the wrong answer with the sum rule, NOT because the rule itself is wrong, but because it is used inappropiatelty. In this case "detecting an electron" is an elementary event, whose probability has to be computed/assigned OUTSIDE of statistics(sum the amplitude and take mod square)
I am not completely sure what do you mean by "evolution of probability law"(not evolution of probability, which is the study of stocastic processes)

If you suggest QM may offer some exciting new angles for statisticians and mathematicians to develope new theories in statistics, I would tend to agree(no doubt someone is already working on that) But I don't know in what direction one should explore.

Last edited: Mar 25, 2005
12. Mar 25, 2005

### bw

One more thing.

Statistics can't tell you whether a system is "truly" random or just behave as though it is(say, because of lack of information) It only offeres tests for sequences that appear to be "sufficiently random", but,a good pseudo random number generator(a completely deterministic machine) can fool a lot of statistics tests. There are many proposals for what may serve as a definition of "true randomness", but none is satisfactory for all concievable occasions as far as I know. There are some interesting attempts to define radomness in terms of complexity theory. But I don't know how they can be actually applied in, say a test.

Therefore we cannot actually say QM is "truly random". We don't know what randomness is!(I am not only saying that non local hidden variables may be at work, the very concept of "randomness" is not well understood, even theoretically!)

13. Mar 25, 2005

### bw

No. "Traditional" probability theory has nothing to say about whether you can
measure quantities simultaneously or not. Again that is a physics question.

Probability theory only says IF the values of these observables are such and such, THEN blah blah...

14. Mar 25, 2005

### SpaceTiger

Staff Emeritus
It has to do with whether the statements of the probability theory make sense for the physical model. Any other interpretation of his question leads to the trivial "of course math works" response. I think he's asking for something deeper than that.

15. Mar 25, 2005

### bw

You mean like a classical phase space but instead of the Liouville flow you have discontinuous flow corresponding to state reduction(?) What information can you gain from that, as you can only "update" after you make a measurement(of course)?

16. Mar 25, 2005

### bw

Actually, now that I think of it I did go to a seminar talk on quantum probability by a mathematician. He was basically trying to incoporate some sort of uncertainty principle type constriants to his system of axioms.I don't know if that led to any interesting result, he didn't seem to claim that he had any new discovery other just setting up the system, But I can't really member the details.

The talk was really bad. The guy got his slides and notes mixed up with a different talk and ended up confusing everyone and the most confused person appeared to be himself. The talk ended in a Sienfeld moment when the guy standing there red faced, hair all messed up with slides all over the bench. We joked that that was the uncertainty principle at work.

Last edited: Mar 25, 2005
17. Mar 25, 2005

### SpaceTiger

Staff Emeritus
Perhaps you could be more specific, as I'm not sure how you intend to describe the combined probability distribution of x and v by including a law for time evolution. Attempts have been made to describe it classically by including non-local "hidden" variables (perhaps these are the extra parameters you're describing), but this is rather ad hoc. On the other hand, that seems ok with you, so I suspect the answer to your question is yes, it can be done. That doesn't mean, however, that it should be done.

18. Mar 26, 2005

### seratend

In the beginning of this thread, even if not correctly expressed (or somewhat fuzzy :), I would like to know if we can distinguish the statistical results of quantum probability (born rules + projection postulate) from the one derived from the Kolgomorovian probability.
The example you gave is correct, but why do you want to mix Kolgomorov probability results with the implicit behaviour of a classical (“Newtonian”) electron? You have implicitly connected the probability slits results at the slits with the probability results at the screen. You have 2 probability laws of 2 random variables. The way you link them is what I call your implicit formal assumption.
Formally, these statistics are not logically connected if you do not add an external assumption (e.g. formally projection postulate, conditional probability, time evolution, etc …).
As you said, we are not considering the time evolution of the system, but rather the evolution of (or the link between) statistics knowing certain results. The projection postulate of QM as well as the conditional probability in Kolgomorov probability include both an implicit formal “evolution” (you can choose a better word) of the statistics (formal view). In Kolgomorv probability you have the basic link P(A=a|B=b) to connect the new probability law to a given result. However, we are free (formally) to build objects that modify the probability space under some experiment results in order to reflect the QM projection postulate of non commuting observables events.

I do not understand completely your statement (as well as I think I am not clear with my statements ; ). A classical probability space is a set of outcomes, a sigma algebra and a probability law. On this space, I may, formally, define any random variable. These random variables (that are not necessarily independent) define the elementary events of experiments based on these random variables (i.e. elementary events are defined respectively to a given random variable). As in QM probability (for a given observable), these elementary events are mutually exclusive for a given random variable (observable). A random variable induces a probability law on its own space as in QM probability. Only the state resulting from the assumption a given event is true seems to be different in QM and Kolgomorov probabilities (what i call the evolution of the law, which is not the best designation).
A is the elementary event “electron in slit A” and B is the elementary event “electron in slit B” for the random variable “electron position at slits” (we may call X_slit)
* (Kolgomorov) P(A)=P(X_slit=xa)= |<slitA|psi>|^2 (QM)
* (Kolgomorov) P(B)=P(X_slit=xb)=|<slitB|psi>|^2 (QM)

We have A inter B = empty set (because we associate them with the random variable X_slit) that is the same as <slitA|slibB>=0 when we consider the observable X_slit (|SlitA> and |SlitB> are eigen values of the observable X_slit).

In this case we have P(AUB)=P(X_slit={xa,xb})=|<slitA|psi>|^2+|<slitB|psi>|^2= P(A)+P(B).

It is the random variable that defines the elementary events in Kolgomorov probability and the observable in QM probability. Formally we have:
* in QM probability (A,|psi>) defines the probability law of the observable A events
* In Kolgomorov probability (A, probability space) defines the probability law of the random variables A events.

Therefore, choosing a probability space (~the law) or a state seems, formally, almost identical to get the same statistical results in an experiment.

I am just trying to see (for one aspect) if QM probabilities may be formally equivalent to Kolgomorov probabilities plus some additional evolution on the probability space (i.e. to match the projection postulate). If this is true, we can view QM probabilities as just a specific case of Kolgomorov probabilities (i.e. the projection postulate may be mapped into an update of the probability space).
Note that the statistics rules are given before the projection of the state in QM probability.

Seratend.

19. Mar 26, 2005

### seratend

Well, I must admit that I do not understand exactly the term random alone.
The only results I know are the results of the strong law of large numbers that allow one to measure the probability law of events on a given experiment (frequentist view of probability).
However, the problem of the strong law of large numbers is that you get a convergence in law. That means, formally when you do an experiment, the peculiar sequence of results you obtain may belong to the events of null probability (i.e. they do not converge to the probability law).
Now, you can say that it is because you have not realized fully "random" trials to explain this result (i.e. if your sequence of trials is not fully random, therefore you may have a sequence of events that does not converge to the probability law). But it seems somewhat artificial.

Seratend.

20. Mar 26, 2005