Amplitudes, Probabilities and EPR

[Jilang]

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.

Sorry, I don't have a registration with that provider. Can the third alternative (fourth-sorry Mike) be summarised here?

 

[Jilang]

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?

Zonde, SD already stressed that it is not the amplitude that gets measured, You don't need to worry about negative clicks.

 

[Nugatory]

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.

Stevendaryl's point about us not having an intuition for what it means to select a result according to an amplitude, as opposed to a probability, is looking pretty good right now...

 

[secur]

@stevendaryl's approach seems to be unaffected by this issue. It still works if we treat the amplitude the normal way, when it comes to selecting an actual result, since that's not crucial in his scheme. I.e., square the amplitude (complex norm) and use that as the probability. Perhaps I'm missing something.

 

[Stephen Tashi]

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.

Can we make the description of the intuitive difficulty more precise?

Mathematically, it is easy to imagine choosing a value according to any sort of input variable. You just need an algorithm that maps values of the input to a value that defines a probability. You can use that probability to make you final choice.

So doesn't the intuitive problem begin in step 4 or 5 instead of in step 1 and 2 ?

 

[mikeyork]

stevendaryl: forgive me if I have misunderstood. But don't we already know what $\lambda$ is? Isn't it the eigenvalue of the composite state? So if $A,B$ are individual spins, then $\lambda$ is the composite spin. And your $\psi(A,B;\alpha,\beta,\lambda)$ are essentially Clebsch-Gordon coefficients -- apart from the rotation which takes the orientation of one detector into the other.

 

[rubi]

Sorry, I don't have a registration with that provider. Can the third alternative (fourth-sorry Mike) be summarised here?

Unfortunately, I don't think it can be understood easily without understanding consistent histories first. The CH answer is that the EPR argument is invalid, because it mixes incompatible frameworks. If you are interested in CH, you should check out Griffiths book "Consistent Quantum Theory". He also has some slides on his homepage: http://quantum.phys.cmu.edu/CHS/histories.html

 

[stevendaryl]

stevendaryl: forgive me if I have misunderstood. But don't we already know what $\lambda$ is? Isn't it the eigenvalue of the composite state? So if $A,B$ are individual spins, then $\lambda$ is the composite spin. And your $\psi(A,B;\alpha,\beta,\lambda)$ are essentially Clebsch-Gordon coefficients -- apart from the rotation which takes the orientation of one detector into the other.

That's on the right track, but not exactly right. In the EPR experiment, the composite spin is zero, so there is only one possible value for that.

No, the meanings of the various amplitudes is this: Let |\Phi\rangle be the composite two-particle spin state. Then

• \lambda = +1 \Rightarrow |\Phi\rangle = |u_z d_z\rangle. The spin state of the first particle (the positron, say) is spin-up in the z-direction, and the spin state of the other particle (the electron) is spin-down in the z-direction.
• \lambda = -1 \Rightarrow |\Phi\rangle = |d_z u_z\rangle. The spin state of the first particle is spin-down in the z-direction, and the spin state of the other particle is spin-up in the z-direction.
• \lambda = +1 \Rightarrow \psi(\lambda) = \frac{1}{\sqrt{2}}
• \lambda = -1 \Rightarrow \psi(\lambda) = \frac{-1}{\sqrt{2}}

So this is just a decomposition of the usual spin-zero state: \frac{1}{\sqrt{2}} (|u_z d_z\rangle - |d_z u_z\rangle). The amplitudes for \lambda = +1 and \lambda = -1 can be read off immediately.

Then the other amplitudes:

\psi_A(A|\alpha, \lambda) = the probability amplitude for measuring spin A/2 in the direction \hat{x} cos(\alpha) + \hat{y} sin(\alpha), given that it was prepared to have spin \lambda/2 in the direction \hat{z}.

\psi_B(B|\alpha, \lambda) = the probability amplitude for measuring spin B/2 in the direction \hat{x} cos(\alpha) + \hat{y} sin(\alpha), given that it was prepared to have spin -\lambda/2 in the direction \hat{z}.

 

[mikeyork]

In the EPR experiment, the composite spin is zero, so there is only one possible value for that.

That's just a special case. Your $\lambda$ can be in any other basis, but I think your $\psi(\lambda)$ functions will be that of a superposition in that new basis equivalent to the composite spin state.

As regards the differing orientations $\alpha$ and $\beta$, that is simply a matter of a frame transformation (a rotation) of the spin projection direction for each detector and that is handled by D-functions (the simplest example being $d(\alpha)$ and $d(\beta)$ -- rotations about the y-axis chosen to be perpendicular to the plane of the z-axis and your direction of projection).

So although you are correct that your $\lambda$ quantum numbers are not simply the composite spin, they are mathematically derived from it.

 

[zonde]

We don't. We have negative (or even complex) amplitudes for positive or zero numbers of clicks.

Stevendaryl's point about us not having an intuition for what it means to select a result according to an amplitude, as opposed to a probability, is looking pretty good right now...

Compare these two expressions:
$P(A,B|\alpha, \beta) = \sum_\lambda P(\lambda) P_A(A|\alpha, \lambda) P_B(B|\beta, \lambda)$ (1)
$\psi(A,B|\alpha, \beta) = \sum_\lambda \psi(\lambda) \psi_A(A|\alpha, \lambda) \psi_B(B|\beta, \lambda)$ (2)
In these two sums we combine non-local summands. There is no problem as these summands are acquired from local values by postprocessing.
The question however is if this summation can be viewed as part of postprocessing. In case of (1) summands are nonnegative and can be viewed as independent elements in postprocessing.
But in (2) summands do not combine statistically as they can cancel each other out so it can't be just a step in postprocessing. So this summation is not consistent with locality.

 

[stevendaryl]

That's just a special case. Your $\lambda$ can be in any other basis, but I think your $\psi(\lambda)$ functions will be that of a superposition in that new basis equivalent to the composite spin state.

As regards the differing orientations $\alpha$ and $\beta$, that is simply a matter of a frame transformation (a rotation) of the spin projection direction for each detector and that is handled by D-functions (the simplest example being $d(\alpha)$ and $d(\beta)$ -- rotations about the y-axis chosen to be perpendicular to the plane of the z-axis and your direction of projection).

Sorry, I misunderstood what you were saying. You're exactly right--the \psi(\lambda) are Clebsch-Gordan coefficients, and the \psi_A(A|\alpha, \beta) are the coefficients relating spin-up in one direction to spin-up in another direction.

 

[stevendaryl]

Compare these two expressions:
$P(A,B|\alpha, \beta) = \sum_\lambda P(\lambda) P_A(A|\alpha, \lambda) P_B(B|\beta, \lambda)$ (1)
$\psi(A,B|\alpha, \beta) = \sum_\lambda \psi(\lambda) \psi_A(A|\alpha, \lambda) \psi_B(B|\beta, \lambda)$ (2)
In these two sums we combine non-local summands. There is no problem as these summands are acquired from local values by postprocessing.
The question however is if this summation can be viewed as part of postprocessing. In case of (1) summands are nonnegative and can be viewed as independent elements in postprocessing.
But in (2) summands do not combine statistically as they can cancel each other out so it can't be just a step in postprocessing. So this summation is not consistent with locality.

I don't understand why you say the sign of the summands implies something about locality, although I do agree that there is something fishy about canceling amplitudes.

Let me go through a pair of "stories", one about probabilities, and one about probability amplitudes, and maybe I can get at the reason that you think there is something weird about amplitudes.

Story 1: Suppose that there is a "left-handed" gene, such that if you have it from either of your parents, you're 81% likely to be left-handed, and if you lack it, you are 81% likely to be right-handed. So a couple has a baby, and by Mendelian genetics, we figure that the baby has a 50% chance of getting the gene from the father. So we compute that he has a 50% chance of being-lefthanded: .5 \cdot .81 + .5 \cdot .19 = .5. Presumably, we could test this empirically by checking many babies in the same genetic situation.

Story 2: Suppose that it works by amplitudes, rather than probabilities. If the baby has the gene, he has an amplitude of 0.9 of being left-handed, and 0.44 of being right-handed. If he lacks the gene, the amplitudes are switched. Now, suppose that we compute that he has an amplitude of +0.44 of having the gene, and -0.9 of lacking the gene. Then the amplitude that he is left-handed is .44 \cdot .9 + (-.9) \cdot .44 = 0. So he has ZERO chance of being left-handed.

The mathematical analysis is very similar in both cases. However, in the first case, each set of parents can reason that really the baby either has the gene, or doesn't, and that the probability reflects their lack of knowledge about the true state of their baby. In the second case, it's hard to see how certainty (that the baby will not be left-handed) can arise from lack of knowledge.

 

[Mentz114]

I don't understand why you say the sign of the summands implies something about locality, although I do agree that there is something fishy about canceling amplitudes.

Let me go through a pair of "stories", one about probabilities, and one about probability amplitudes, and maybe I can get at the reason that you think there is something weird about amplitudes.
..
..
The mathematical analysis is very similar in both cases. However, in the first case, each set of parents can reason that really the baby either has the gene, or doesn't, and that the probability reflects their lack of knowledge about the true state of their baby. In the second case, it's hard to see how certainty (that the baby will not be left-handed) can arise from lack of knowledge.

It looks like you've introduced interference which rules out one possibility. Lack of information is easy to define in quantum systems but (apparently) has a different interpretation between CM and QM. I found this paper very interesting on this subject

arXiv:quant-ph/9601025v1 25 Jan 1996

 

[zonde]

I don't understand why you say the sign of the summands implies something about locality, although I do agree that there is something fishy about canceling amplitudes.

Let me go through a pair of "stories", one about probabilities, and one about probability amplitudes, and maybe I can get at the reason that you think there is something weird about amplitudes.

Story 1: Suppose that there is a "left-handed" gene, such that if you have it from either of your parents, you're 81% likely to be left-handed, and if you lack it, you are 81% likely to be right-handed. So a couple has a baby, and by Mendelian genetics, we figure that the baby has a 50% chance of getting the gene from the father. So we compute that he has a 50% chance of being-lefthanded: .5 \cdot .81 + .5 \cdot .19 = .5. Presumably, we could test this empirically by checking many babies in the same genetic situation.

Story 2: Suppose that it works by amplitudes, rather than probabilities. If the baby has the gene, he has an amplitude of 0.9 of being left-handed, and 0.44 of being right-handed. If he lacks the gene, the amplitudes are switched. Now, suppose that we compute that he has an amplitude of +0.44 of having the gene, and -0.9 of lacking the gene. Then the amplitude that he is left-handed is .44 \cdot .9 + (-.9) \cdot .44 = 0. So he has ZERO chance of being left-handed.

The mathematical analysis is very similar in both cases. However, in the first case, each set of parents can reason that really the baby either has the gene, or doesn't, and that the probability reflects their lack of knowledge about the true state of their baby. In the second case, it's hard to see how certainty (that the baby will not be left-handed) can arise from lack of knowledge.

Your Story 2 is quite counterintuitive but it's not the problem I see there. It is a step further where I see the problem.
In your description of entangled pair, amplitudes are attached to different coincidences not simply local measurements (as in your example with babies). And then amplitudes of coincidences can cancel out. The problem is that coincidences are not basic measurements but rather derived by postselection. So it would seem that this cancelation should happen somewhere in the process of postselection.
Imagine experimentalist who compares two sheets of paper where there are "clicks" of detectors with timestamps from two locations. He then on the third paper counts coincidences in two columns as being the same polarization or opposite polarization. But then he sometimes subtracts coincidence from one or the other column (amplitudes canceling out). That is the strange part in it.

 

[stevendaryl]

Your Story 2 is quite counterintuitive but it's not the problem I see there. It is a step further where I see the problem.
In your description of entangled pair, amplitudes are attached to different coincidences not simply local measurements (as in your example with babies). And then amplitudes of coincidences can cancel out. The problem is that coincidences are not basic measurements but rather derived by postselection. So it would seem that this cancelation should happen somewhere in the process of postselection.
Imagine experimentalist who compares two sheets of paper where there are "clicks" of detectors with timestamps from two locations. He then on the third paper counts coincidences in two columns as being the same polarization or opposite polarization. But then he sometimes subtracts coincidence from one or the other column (amplitudes canceling out). That is the strange part in it.

The summation \sum_\lambda \psi(\lambda) \psi_A(A|\alpha, \lambda) \psi_B(B|\beta, \lambda) does not represent any kind of post-processing, because \lambda is hidden, so nobody ever knows what its value was. The meaning of that sum is that this is a model explaining how the amplitude \psi(A, B|\alpha,\beta) might arise. So I'm not sure I understand why you say it's nonlocal.

I guess in general, it's sort of nonlocal to add probabilities, or probability amplitudes, but that seems like a necessary step in order to talk about nonlocal correlations, which is what hidden variables are supposed to explain. I don't understand why you think it's more nonlocal to talk about amplitudes than to talk about probabilities.

 

[zonde]

The summation \sum_\lambda \psi(\lambda) \psi_A(A|\alpha, \lambda) \psi_B(B|\beta, \lambda) does not represent any kind of post-processing, because \lambda is hidden, so nobody ever knows what its value was. The meaning of that sum is that this is a model explaining how the amplitude \psi(A, B|\alpha,\beta) might arise. So I'm not sure I understand why you say it's nonlocal.

I guess in general, it's sort of nonlocal to add probabilities, or probability amplitudes, but that seems like a necessary step in order to talk about nonlocal correlations, which is what hidden variables are supposed to explain. I don't understand why you think it's more nonlocal to talk about amplitudes than to talk about probabilities.

Let me ask a bit more about role of $\lambda$ in your model. Do your model assumes that each detection of pair has only one of the possible values of $\lambda$ attached to it? Or does every observation of pair have both values of $\lambda$ attached to it?
In other words do we get averaged amplitude \psi(A, B|\alpha,\beta) for every coincidence or do we get it only when we average over ensemble of pairs?

 

[stevendaryl]

Let me ask a bit more about role of $\lambda$ in your model. Do your model assumes that each detection of pair has only one of the possible values of $\lambda$ attached to it? Or does every observation of pair have both values of $\lambda$ attached to it?
In other words do we get averaged amplitude \psi(A, B|\alpha,\beta) for every coincidence or do we get it only when we average over ensemble of pairs?

My description in terms of \lambda is just an alternative way to describe the standard quantum mechanical computation of transition amplitudes, to make the description more parallel to a classical hidden-variables probabilistic model.

The traditional approach is this:

• Let |\phi_\lambda\rangle be a complete basis for the system of interest.
• Let H be its Hamiltonian.
  
• Let O_\alpha be a set of possible observables.
• Let |\chi_{\alpha, j}\rangle be a different complete basis in which the operator O_\alpha is diagonal. Let a_{\alpha, j} be the eigenvalue of O_\alpha for state |\chi_{\alpha, j}\rangle. That is, O_\alpha |\chi_{\alpha, j}\rangle = a_{\alpha, j} |\chi_{\alpha, j}\rangle. For simplicity, let's assume that there is no degeneracy; there is only state |\chi_{\alpha, j}\rangle with the eigenvalue a_{\alpha,j}.
  
• Let \gamma_{\alpha, a, \lambda}(t) be the transition amplitude: \langle \chi_{\alpha,j} | e^{-iHt} |\phi_\lambda \rangle, where j is the unique index such that a_{\alpha, j} = a

With all that, we can compute probability amplitudes for getting certain measurement results from certain initial states (Fix the time t that the measurement will be performed, for simplicity):

• Write the initial state |\psi\rangle (at time t=0) as a superposition of basis states: |\psi\rangle = \sum_\lambda c_\lambda |\phi_\lambda\rangle
• Then if the observer measures observable O_\alpha at time t, the probability amplitude of measuring value a is given by: \sum_\lambda c_\lambda \gamma_{\alpha,a,\lambda}(t)

The translation to the "hidden variables" approach is trivial:

• Let the hidden variable, \lambda, be the index of the complete basis used to describe the initial state.
  
• Let the initial amplitude, \psi(\lambda) be given by \psi(\lambda) = c_\lambda, the coefficient of the basis element |\phi_\lambda of the initial state.
• Let the conditional amplitude be given by \psi(a | \alpha, \lambda) = \gamma_{\alpha,a,\lambda}(t).
• Then the amplitude for getting result a at time t is given by: \psi_a(a | \alpha) = \sum_\lambda \psi(\lambda) \psi(a | \alpha, \lambda)

The EPR case is only more complicated, in that the observable being measured is a pair of values, one measured by Alice and one measured by Bob, and the initial state is a product state.

 

[zonde]

The EPR case is only more complicated, in that the observable being measured is a pair of values, one measured by Alice and one measured by Bob, and the initial state is a product state.

For your model to work in EPR case you would have to model post-processing using amplitudes i.e. detection results would have to remain hidden variables until coincidences are obtained and "measured".

 

[stevendaryl]

For your model to work in EPR case you would have to model post-processing using amplitudes i.e. detection results would have to remain hidden variables until coincidences are obtained and "measured".

I don't understand your point about "post-processing", but I think I've said everything that there is to say: The goal was simply to write the amplitude for joint measurements as an amplitude-weighted sum of uncorrelated (product) amplitudes. It's just a mathematical exercise---I'm not making any claims about having any new interpretation of QM. It's just a way of describing the usual QM that parallels what is done with hidden-variables models (except that probabilities are replaced by amplitudes).

 

[RockyMarciano]

For your model to work in EPR case you would have to model post-processing using amplitudes i.e. detection results would have to remain hidden variables until coincidences are obtained and "measured".

I think you are under a similar confusion that secur initially had concerning what the OP model intends. If it "worked" in the way that you seem to think it would it would be a counterexample to Bell's theorem and that is not at all its aim.

 

[RockyMarciano]

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.

.
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.

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.

I do think these observations might contribute to a better understanding in that they analyze a specific quantum situation in which it is very clear that the amplitudes cannot correspond to measurements, given the bipartite system prepared as pure states used, so only the probabilities are relevant, so while it is clean enough that can never raise any doubt about Bell's theorem it gives us hints about how complex amplitudes being squared erase any trace of hidden variables in QM and therefore helps clarify the mathematical device that the formulism uses to achieve this.

 

[secur]

The only problem (as I indicated before) is that using this approach, you can make it result in the correct probabilities, namely, 1/2 cos^2 and sin^2 of (\alpha - \beta)/2. That seems to undermine the conclusion that "The squaring process is in some sense responsible for the weirdness of QM correlations".

 

[RockyMarciano]

The only problem (as I indicated before) is that using this approach, you can make it result in the correct probabilities, namely, 1/2 cos^2 and sin^2 of (\alpha - \beta)/2. That seems to undermine the conclusion that "The squaring process is in some sense responsible for the weirdness of QM correlations".

That sentence may be ambiguous in its meaning, I think that rather than making certain operation responsible for any perceived weirdness, what is shown is how the formalism (Born rule)attains nonlocal correlations from local amplitudes
not connected to measurements directly. It is of course nature that is responsible for the lack of predetermined values of measurements, not any mathematical operation.

 

[lavinia]

The Shroedinger Equation for a free particle is just the Heat Equation with an $i$ thrown in. It is no surprise that it should behave like a complex diffusion - meaning that conditional amplitudes replace conditional probabilities.

 

[RockyMarciano]

The Shroedinger Equation for a free particle is just the Heat Equation with an $i$ thrown in. It is no surprise that it should behave like a complex diffusion - meaning that conditional amplitudes replace conditional probabilities.

This is indeed no surprise, but the OP shows a little more than this. At least I thought that what it was highlighting by using the amplitudes obtained from the squared root of the probabilities(i.e. going backwards with respect to the usual process from complex amplitudes to probabilities) and therefore up to phase, which led to a local hidden variables model of those amplitudes, was the contrast with the usual forward procedure from complex amplitudes being multiplied to their complex conjugates and where some info about phase makes it to the probabilities. This can't happen with classical probabilities and real valued inputs for obvious mathematical reasons.

 

[lavinia]

One can derive the Shroedinger equation from the assumption that the process of state evolution is like a Markov process except with conditional probabilities replaced by conditional amplitudes.

 

[Jilang]

One can derive the Shroedinger equation from the assumption that the process of state evolution is like a Markov process except with conditional probabilities replaced by conditional amplitudes.

That's interesting. Do you have link for that?

 

[lavinia]

That's interesting. Do you have link for that?

Feynmann's Lecture Volume 3. There is another post where I wrote out what Feynman said. Forget what it was called.

 

[Jilang]

Feynmann's Lecture Volume 3. There is another post where I wrote out what Feynman said. Forget what it was called.

Thanks but I was meaning the derivation itself.

 

[Stephen Tashi]

In a non-QM context, complex numbers have been used in the place of real number transition probabilites in Markov chains. The simplest example, I know of this is the paper : http://bidabad.com/doc/complex-prob.pdf. The general idea is that the we have data for a Markov process who steps occur increments of time T and we wish to have a model that proceeds in smaller steps of time or a model that is a continuous time Markov process.

I don't know what approach this paper takes, but it's an oft-cited work:
D.R. Cox, A use of Complex Probabilities in the Theory of Stochastic Processess
https://www.cambridge.org/core/jour...processesdiv/3DE2C9013903EDD218F5B85129F65B2C

I don't subscribe to that site and I haven't been able to find a free article that gives that citation and also explains the concept of the paper.