quantumapplitudes

Quantum Amplitudes, Probabilities and EPR

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This is a little note about quantum amplitudes. Even though quantum probabilities seem very mysterious, with weird interference effects and seemingly nonlocal effects, the mathematics of quantum amplitudes are completely straight-forward. (The amplitude squared gives the probability.) As a matter of fact, the rules for computing amplitudes are almost exactly the same as the classical rules for computing probabilities for a memoryless stochastic process. (Memoryless means that future probabilities depend only on the current state, not on how it got to that state.)

Probabilities for stochastic processes:

If you have a stochastic process such as Brownian motion, then probabilities work this way:

Let [itex]P(i,t|j,t’)[/itex] be the probability that the system winds up in state [itex]i[/itex] at time [itex]t[/itex], given that it is in state [itex]j[/itex] at time [itex]t'[/itex].

Then these transition probabilities combine as follows: (Assume [itex]t’ < t” < t[/itex])

[itex]P(i,t|j,t’) = \sum_k P(i,t|k,t”) P(k,t”|j,t’)[/itex]

where the sum is over all possible intermediate states [itex]k[/itex].

There are two principles at work here:

  1. In computing the probability for going from state [itex]j[/itex] to state [itex]k[/itex] to state [itex]i[/itex], you multiply the probabilities for each “leg” of the path.
  2. In computing the probability for going from state [itex]j[/itex] to state [itex]i[/itex] via an intermediate state, you add the probabilities for each alternative intermediate state.

These are exactly the same two rules for computing transition amplitudes using Feynman path integrals. So there is an analogy: amplitudes are to quantum mechanics as probabilities are to classical stochastic processes.

Continuing with the analogy, we can ask the question as to whether there is a local hidden variables theory for quantum amplitudes. The answer is YES.

Local “hidden-variables” model for EPR amplitudes

Here’s a “hidden-variables” theory for the amplitudes for the EPR experiment.

First, a refresher on the probabilities for the spin-1/2 anti-correlated EPR experiment, and what a “hidden-variables” explanation for those probabilities would be:

In the EPR experiment, there is a source for anti-correlated electron-positron pairs. One particle of each pair is sent to Alice, and another is sent to Bob. They each measure the spin relative to some axis that they choose independently.

Assume Alice chooses her axis at angle [itex]\alpha[/itex] relative to the x-axis in the x-y plane, and Bob chooses his to be at angle [itex]\beta[/itex] (let’s confine the orientations of the detectors to the x-y plane, so that orientation can be given by a single real number, an angle). Then the prediction of quantum mechanics is that probability that Alice will get result [itex]A[/itex] (+1 for spin-up, relative to the detector orientation, and -1 for spin-down) and Bob will get result [itex]B[/itex] is:

[itex]P(A, B | \alpha, \beta) = \frac{1}{2} sin^2(\frac{\beta-\alpha}{2}) [/itex] if [itex]A = B[/itex]
[itex]P(A, B | \alpha, \beta) = \frac{1}{2} cos^2(\frac{\beta-\alpha}{2}) [/itex] if [itex]A \neq B[/itex]

A “local hidden variables” explanation for this result would be given by a probability distribution [itex]P(\lambda)[/itex] on values of some hidden variable [itex]\lambda[/itex], together with probability distributions

[itex]P_A(A | \alpha, \lambda)[/itex]
[itex]P_B(B | \beta, \lambda)[/itex]

such that

[itex]P(A, B | \alpha, \beta) = \sum P(\lambda) P_A(A|\alpha, \lambda) P_B(B|\beta, \lambda)[/itex]

(where the sum is over all possible values of [itex]\lambda[/itex]; if [itex]\lambda[/itex] is continuous, the sum should be replaced by [itex]\int d\lambda[/itex].)

The fact that the QM predictions violate Bell’s inequality proves that there is no such hidden-variables explanation of this sort.

But now, let’s go through the same exercise in terms of amplitudes, instead of probabilities. The amplitude for Alice and Bob to get their respective results is basically the square-root of the probability (up to a phase). So let’s consider the amplitude:

[itex]\psi(A, B|\alpha, \beta) \sim \frac{1}{\sqrt{2}} sin(\frac{\beta – \alpha}{2})[/itex] if [itex]A = B[/itex], and
[itex]\psi(A, B|\alpha, \beta) \sim \frac{1}{\sqrt{2}} cos(\frac{\beta – \alpha}{2})[/itex] if [itex]A \neq B[/itex].

(I’m using the symbol [itex]\sim[/itex] to mean “equal up to a phase”; I’ll figure out a convenient phase as I go).

In analogy with the case for probabilities, let’s say a “hidden variables” explanation for these amplitudes will be a parameter [itex]\lambda[/itex] with associated functions [itex]\psi(\lambda)[/itex], [itex]\psi_A(A|\lambda, \alpha)[/itex], and [itex]\psi_B(B|\lambda, \beta)[/itex] such that:

[itex]\psi(A, B|\alpha, \beta) = \sum \psi(\lambda) \psi_A(A | \alpha, \lambda) \psi_B(B | \beta, \lambda)[/itex]

where the sum ranges over all possible values for the hidden variable [itex]\lambda[/itex].
I’m not going to bore you (any more than you are already) by deriving such a model, but I will just present it:

  1. The parameter [itex]\lambda[/itex] ranges over the two-element set, [itex]\{ +1, -1 \}[/itex]
  2. The amplitudes associated with these are: [itex]\psi(\lambda) = \frac{\lambda}{\sqrt{2}} = \pm \frac{1}{\sqrt{2}}[/itex]
  3. When [itex]\lambda = +1[/itex], [itex]\psi_A(A | \alpha, \lambda) = A \frac{1}{\sqrt{2}} e^{i \alpha/2}[/itex] and [itex]\psi_B(B | \beta, \lambda) = \frac{1}{\sqrt{2}} e^{-i \beta/2}[/itex]
  4. When [itex]\lambda = -1[/itex], [itex]\psi_A(A | \alpha, \lambda) = \frac{1}{\sqrt{2}} e^{-i \alpha/2}[/itex] and [itex]\psi_B(B | \alpha, \lambda) = B \frac{1}{\sqrt{2}} e^{i \beta/2}[/itex]

Check:
[itex]\sum \psi(\lambda) \psi_A(A|\alpha, \lambda) \psi_B(B|\beta, \lambda) = \frac{1}{\sqrt{2}} (A \frac{1}{\sqrt{2}} e^{i \alpha/2}\frac{1}{\sqrt{2}} e^{-i \beta/2} – \frac{1}{\sqrt{2}} e^{-i \alpha/2} B \frac{1}{\sqrt{2}} e^{+i \beta/2})[/itex]

If [itex]A = B = \pm 1[/itex], then this becomes (using [itex]sin(\theta) = \frac{e^{i \theta} – e^{-i \theta}}{2i}[/itex]):

[itex] = \pm 1 \frac{i}{\sqrt{2}} sin(\frac{\alpha – \beta}{2})[/itex]

If [itex]A = -B = \pm 1[/itex], then this becomes (using [itex]cos(\theta) = \frac{e^{i \theta} + e^{-i \theta}}{2}[/itex]):

[itex] = \pm 1 \frac{1}{\sqrt{2}} cos(\frac{\alpha – \beta}{2})[/itex]

So we have successfully reproduced the quantum predictions for amplitudes (up to the phase [itex]\pm 1[/itex]).

What does it mean?
 
In a certain sense, what this suggests is that quantum mechanics is a sort of “stochastic process”, but where the “measure” of possible outcomes of a transition is not real-valued probabilities but complex-valued probability amplitudes. When we just look in terms of amplitudes, everything seems to work out the same as it does classically, and the weird correlations that we see in experiments such as EPR are easily explained by local hidden variables, just as Einstein, Podolsky and Rosen hoped. 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.

Do these observations contribute anything to our understanding of QM? Beats me. But they are interesting.

 

 

83 replies
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  1. mikeyork
    mikeyork says:

    [QUOTE="stevendaryl, post: 5634793, member: 372855"]amplitudes don't correspond directly to anything can measure[/QUOTE]

    Because of an arbitrary global phase. But relative amplitudes are physically significant. Spin-statistics is an obvious example.

  2. mfb
    mfb says:

    [QUOTE="mikeyork, post: 5634970, member: 22888"]Because of an arbitrary global phase. But relative amplitudes are physically significant. Spin-statistics is an obvious example.[/QUOTE]It looks like you talk about phases, not amplitudes.

  3. secur
    secur says:

    [QUOTE="stevendaryl, post: 5634793, member: 372855"]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.[/QUOTE]

    Yes it's interesting, and now it's a lot clearer to me, as delineated in my …[QUOTE="secur, post: 5634486, member: 588176"]previous post.[/QUOTE]

    Initially I thought you were talking about the standard Bell experiment hidden-variables situation. You're actually doing something a bit different, and unique, AFAIK. That's a problem with an original idea: many people will mistake it for the "same old thing" they've heard before. Reviewing OP the misunderstanding seems pretty natural. On the plus side we've been able to bring out some of the subtleties of the standard Bell HV model, by contrast. Also, no doubt others thought the same as I did, so it was worthwhile to straighten that out. Thanks for the stimulating thread!

  4. stevendaryl
    stevendaryl says:

    Just to expand a little bit about the analogy between classical probabilities and quantum amplitudes:

    Here is the (false, as shown by tests of Bell's inequality) classical nondeterministic local hidden-variables story for correlated measurements:

    1. There is a source of twin particles. Each twin particle is associated with a hidden parameter, [itex]lambda[/itex]. The source randomly chooses a value of [itex]lambda[/itex] according to some probability distribution [itex]P(lambda)[/itex].
    2. Alice chooses a detector setting [itex]alpha[/itex] for her measurement. Her detector randomly chooses a value for the output, [itex]A[/itex], according to a probability distribution [itex]P_A(A|alpha, lambda)[/itex], which depends on both [itex]lambda[/itex] and [itex]alpha[/itex].
    3. Bob chooses a detector setting [itex]beta[/itex] for his measurement. His detector randomly chooses a value for the output, [itex]B[/itex], according to a probability distribution [itex]P_B(B|beta, lambda)[/itex], which depends on both [itex]lambda[/itex] and [itex]beta[/itex].
    4. Steps 2&3 are independent, so the joint probability is just a product of the individual probabilities: [itex]P(A,B|alpha, beta, lambda) = P_A(A|alpha, lambda) P_B(B|beta, lambda)[/itex].
    5. Since Alice and Bob don't know [itex]lambda[/itex], we average over all possible values, weighted by the probability [itex]P(lambda)[/itex], to get a correlated joint probability distribution: [itex]P(A,B|alpha, beta) = sum_lambda P(lambda) P(A,B|alpha, beta, lambda)[/itex]

    So this story would explain the correlation in Alice's and Bob's measurements as being due to a common (though unknown) hidden variable, [itex]lambda[/itex]. That's what a local hidden variable theory would do, if there were one. Note, that even though this model is nondeterministic, all choices being made—which value of [itex]lambda[/itex], which value of [itex]A[/itex], which value of [itex]B[/itex]—are made using only local information.

    Here's the analogous story for amplitudes:

    1. There is a source of twin particles. Each twin particle is associated with a hidden parameter, [itex]lambda[/itex]. The source randomly chooses a value of [itex]lambda[/itex] according to some amplitude [itex]psi(lambda)[/itex].
    2. Alice chooses a detector setting [itex]alpha[/itex] for her measurement. Her detector randomly chooses a value for the output, [itex]A[/itex], according to an amplitude [itex]psi_A(A|alpha, lambda)[/itex], which depends on both [itex]lambda[/itex] and [itex]alpha[/itex].
    3. Bob chooses a detector setting [itex]beta[/itex] for his measurement. His detector randomly chooses a value for the output, [itex]B[/itex], according to an amplitude [itex]psi_B(B|beta, lambda)[/itex], which depends on both [itex]lambda[/itex] and [itex]beta[/itex].
    4. Steps 2&3 are independent, so the joint amplitude is just a product of the individual probabilities: [itex]psi(A,B|alpha, beta, lambda) = psi_A(A|alpha, lambda) psi_B(B|beta, lambda)[/itex].
    5. Since Alice and Bob don't know [itex]lambda[/itex], we average over all possible values, weighted by the amplitude [itex]psi(lambda)[/itex] to get a correlated joint probability distribution: [itex]psi(A,B|alpha, beta) = sum_lambda psi(lambda) psi(A,B|alpha, beta, lambda)[/itex]

    The amplitude story has one final step:

    6. We compute a probability from the amplitude, according to the rule [itex]P(A,B|alpha, beta) = |psi(A,B|alpha,beta)|^2[/itex]

    The hidden-variables amplitude story sounds as local as the hidden-variables probability story. 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.

    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.

  5. rubi
    rubi says:

    [QUOTE="stevendaryl, post: 5635253, member: 372855"]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.[/QUOTE]

    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.

  6. zonde
    zonde says:

    [QUOTE="stevendaryl, post: 5633728, member: 372855"]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.[/QUOTE]

    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?

  7. Jilang
    Jilang says:

    [QUOTE="rubi, post: 5635291, member: 395236"]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.[/QUOTE]

    Without hidden variables or interactions over space time distances what is another explanation?

  8. mikeyork
    mikeyork says:

    [QUOTE="Jilang, post: 5635356, member: 492883"]Without hidden variables or interactions over space time distances what is another explanation?[/QUOTE]

    Here's one: space-time distances are a creation of the observer, not fundamental to reality. We already know from relativity that space-time is observer-dependent. So what is it in the absence of an observer?

  9. rubi
    rubi says:

    [QUOTE="Jilang, post: 5635356, member: 492883"]Without hidden variables or interactions over space time distances what is another explanation?[/QUOTE]

    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/55/1/10.1119/1.14965

    [Mentor's note:  This post has been edited to remove a reply to a deleted post]

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