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Please read this Insight here:
https://www.physicsforums.com/insights/quantum-amplitudes-probabilities-epr/
https://www.physicsforums.com/insights/quantum-amplitudes-probabilities-epr/
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I think yes. There are two diverse concepts of probability. The primary one that we think of all the time is that which corresponds to frequency counting. It can be determined only asymptotically as the number of possible events approaches infinity. But there is also a theoretical concept of probability which we use all the time before even a single event is detected. Historically we try to predict probability as that asymptotic limit itself. But, any quantity from which the asymptotic limit is uniquely computable is an encoding of theoretical probability. In QM the amplitude is such an encoding. As such a probability encoding it enables the probabilistic prediction of quantum states without detecting any events. It has all the requirements of a theoretical probability encoding while at the same time describing the physical reality of a single system. Frequency counting, on the other hand, has meaning only for large numbers of systems/events. We can think of QM and the probability amplitude, therefore, as the primary encoding of theoretical probability and frequency counting as secondary.stevendaryl said:Do these observations contribute anything to our understanding of QM?
We can talk about probabilities when assumption that ensemble is i.i.d. holds. When this assumption holds probabilities of individual events independently contribute to total probability. Amplitudes obviously are not independent as two opposite amplitudes can cancel out.stevendaryl said:Do these observations contribute anything to our understanding of QM? Beats me. But they are interesting.
zonde said:We can talk about probabilities when assumption that ensemble is i.i.d. holds. When this assumption holds probabilities of individual events independently contribute to total probability. Amplitudes obviously are not independent as two opposite amplitudes can cancel out.
It is interesting that in your model the final amplitude is calculated by adding/subtracting amplitudes from two different subsets of pairs (with ##\lambda## +1 and -1). So each separate pair does not produce correct amplitude. And they add up not just statistically but in a way that suggest some interdependence on the level of ensemble of pairs.
zonde said:We can talk about probabilities when assumption that ensemble is i.i.d. holds.
Yes, that's true! A similar conclusion is also drawn instevendaryl said:But in actually testing the predictions of quantum mechanics, we can't directly measure amplitudes, but instead compile statistics which give us probabilities, which are the squares of the amplitudes. The squaring process is in some sense responsible for the weirdness of QM correlations.
Independent and identically distributedstevendaryl said:I skipped over this first line without asking: What does "i.i.d." stands for?
stevendaryl said:Amplitudes add in the same way that probabilities do. The reason that some amplitudes cancel others is because they aren't guaranteed to be positive.
secur said:Sorry, this is not a hidden-variable model as understood in Bell experiment. The problem is that A and B results must be +-1. When you calculate
[itex]\psi(A, B|\alpha, \beta) = \sum \psi(\lambda) \psi_A(A | \alpha, \lambda) \psi_B(B | \beta, \lambda)[/itex]
the numbers that appear for [itex] \psi_A(A | \alpha, \lambda)[/itex] and [itex]\psi_B(B | \beta, \lambda)[/itex] can't be complex; can't even be any real numbers, except +-1. That's because this calculation must be performed on their actual results, after the experiment is concluded.
secur said:@Jilang. the wavefunctions can, in fact, be complex. I said that in the calculation of [itex]\psi(A, B|\alpha, \beta)[/itex] "the numbers that appear for [itex] \psi_A(A | \alpha, \lambda)[/itex] and [itex]\psi_B(B | \beta, \lambda)[/itex] can't be complex".
secur said:Sorry, this is not a hidden-variable model as understood in Bell experiment. The problem is that A and B results must be +-1 ...
... your scheme doesn't guarantee that if [itex]\alpha[/itex] = [itex]\beta[/itex] then their results will definitely be opposite. A and B must "flip a coin" based on their amplitudes, and record a definite +1 or -1. In general with these probability amplitudes they will often get the same result, even though the angles are equal.
secur said:Seems there's some confusion. I think the best way to straighten it out is, please address the other point I made, which is very simple.
In this typical Bell-type experiment, QM says A and B must always be opposite (product is -1) when their detector angles are equal. A valid hidden-variable model must reproduce that behavior. But that's not the case with your model:
stevendaryl said:It certainly is. The resulting probability amplitude is (the very first post):
•If [itex]A=B=\pm 1[/itex], then [itex]\psi(A, B|\alpha, \beta) = \pm \frac{i}{\sqrt{2}} sin(\frac{\beta-\alpha}{2})[/itex]. This means that the probability amplitude that Alice and Bob both get the same result is proportional to [itex]sin(\frac{\beta-\alpha}{2})[/itex], which means it is zero when [itex]\alpha = \beta[/itex].
•If [itex]A=-B=\pm 1[/itex], then [itex]\psi(A, B|\alpha, \beta) = \pm \frac{1}{\sqrt{2}} cos(\frac{\beta-\alpha}{2})[/itex]. This means that the probability amplitude that Alice and Bob get opposite results is proportional to [itex]cos(\frac{\beta - \alpha}{2})[/itex], which means that it's 0 when [itex]\alpha - \beta = \pi[/itex].
stevendaryl said:Look, the whole point of the first post was to reproduce the EPR spin-1/2 joint probability function.
stevendaryl said:The fact that the QM predictions violate Bell's inequality proves that there is no such hidden-variables explanation of this sort.
stevendaryl said:The squaring process is in some sense responsible for the weirdness of QM correlations.
secur said:Ok, I thought that answer would remove my confusion. This is NOT a hidden-variables model of the type addressed by Bell's theorem.
Because of an arbitrary global phase. But relative amplitudes are physically significant. Spin-statistics is an obvious example.stevendaryl said:amplitudes don't correspond directly to anything can measure
It looks like you talk about phases, not amplitudes.mikeyork said:Because of an arbitrary global phase. But relative amplitudes are physically significant. Spin-statistics is an obvious example.
stevendaryl said:The point, which I made in the very first post, is that
1.We can formulate certain mathematical rules for how we think that probability ought to work, in a local realistic model.
2.We can prove that QM probabilities don't work that way.
3.However, the analogous rules for QM amplitudes do work that way.
Amplitudes work for QM in the way that we would expect probabilities to work in a local hidden variables model of the sort Bell investigated. As you say, and as I said in the very first post, amplitudes don't correspond directly to anything can measure, unlike probabilities, so it's unclear what relevance this observation is. I just thought it was interesting.
secur said:previous post.
Yes, of course; the magnitude is already physically significant in giving the probability.mfb said:It looks like you talk about phases, not amplitudes.
I don't think that this is how people argue for locality of QM. The argument for locality is that a hidden parameter is not the only possible explanation for the correlations, because mathematically, the assumption of a hidden parameter is a non-trivial restriction on the set of models (i.e. hidden variable models aren't the most general models). In order for a particular model to be local, that model just needs to offer an explanation for how the correlations can come about without invoking interactions over space-like distances.stevendaryl said:And as a matter of fact, when people give rigorous mathematical proofs of the locality of quantum mechanics or quantum field theory, they are really showing that amplitudes behave locally, even if probabilities do not.
Let's say I am giving you apples. Every time I give you apples we describe this event with positive (or at least non negative) integer. Every such event can be viewed as independent because it's different apples every time. But now let's say that event of me giving you apples can be described by any integer (positive, negative or zero). If I give you negative number of apples it actually means I am taking apples from you. Obviously event of taking away apples is not independent from event of giving you apples as the same apples participate in both events.stevendaryl said:Amplitudes add in the same way that probabilities do. The reason that some amplitudes cancel others is because they aren't guaranteed to be positive.
Without hidden variables or interactions over space time distances what is another explanation?rubi said:I don't think that this is how people argue for locality of QM. The argument for locality is that a hidden parameter is not the only possible explanation for the correlations, because mathematically, the assumption of a hidden parameter is a non-trivial restriction on the set of models (i.e. hidden variable models aren't the most general models). In order for a particular model to be local, that model just needs to offer an explanation for how the correlations can come about without invoking interactions over space-like distances.
That depends on the model. There are several manifestly local quantum mechanical models. One example would be consistent histories. A careful analysis of the EPR paradox is done in the following paper:Jilang said:Without hidden variables or interactions over space time distances what is another explanation?
Sorry, I don't have a registration with that provider. Can the third alternative (fourth-sorry Mike) be summarised here?rubi said:That depends on the model. There are several manifestly local quantum mechanical models. One example would be consistent histories. A careful analysis of the EPR paradox is done in the following paper:
http://scitation.aip.org/content/aapt/journal/ajp/55/1/10.1119/1.14965Space-time is not observer dependent. Relativity doesn't claim that.
Zonde, SD already stressed that it is not the amplitude that gets measured, You don't need to worry about negative clicks.zonde said:Let's say I am giving you apples. Every time I give you apples we describe this event with positive (or at least non negative) integer. Every such event can be viewed as independent because it's different apples every time. But now let's say that event of me giving you apples can be described by any integer (positive, negative or zero). If I give you negative number of apples it actually means I am taking apples from you. Obviously event of taking away apples is not independent from event of giving you apples as the same apples participate in both events.
But how would you model "negative" click in detector?
We don't. We have negative (or even complex) amplitudes for positive or zero numbers of clicks.zonde said:But how would you model "negative" click in detector?
stevendaryl said:The screwy thing about the amplitude story is that we have an intuitive idea about what it means to choose a value according to a certain probability distribution (rolling dice, for instance), but we don't have an intuitive idea about what it means to choose a value according to a certain amplitude.