EPR in Bohm formulation

[cianfa72]

TL;DR

EPR paradox in Bohm simplified formulation

Hi, I was reading about the EPR paradox in Bohm simplified formulation.

From my understanding the paradox is that Bob is actually able to get a value for the positron's spin along both the $z$ and $x$ axes.

Since electron and positron are entangled, he get the value of spin along $z$ axis from the Alice's measurement along $z$-axis and the spin along the $x$-axis by its own measurement.

Is the above correct? Thanks.

 

[PeterDonis]

From my understanding the paradox is that Bob is actually able to get a value for the positron's spin along both the $z$ and $x$ axes.

No. The claimed "paradox" is that, if EPR are right about "elements of reality", Bob should be able to do this: but according to QM, he can't. EPR claimed that this meant QM had to be incomplete.

However, we now know that EPR were simply wrong about "elements of reality": if they were right, no experiment would show any violations of the Bell inequalities. But experiments do show such violations.

 

[cianfa72]

No. The claimed "paradox" is that, if EPR are right about "elements of reality", Bob should be able to do this: but according to QM, he can't. EPR claimed that this meant QM had to be incomplete.

He can't since the measurement spin operators along $z$ axis and along $x$ axis for the entire entangled system actually do not commute, right ?

 

[PeterDonis]

He can't since the measurement spin operators along $z$ axis and along $x$ axis for the entire entangled system actually do not commute, right ?

The operators for each particle individually do not commute. QM says that since the operators do not commute, it is impossible for the system they are operating on to have definite values for both. So it is impossible for Bob to have a definite value for both $z$ spin and $x$ spin for his particle.

 

[cianfa72]

So it is impossible for Bob to have a definite value for both $z$ spin and $x$ spin for his particle.

Sorry, maybe I misinterpreted the paradox.

The point is that, starting from Alice spin measurement on her particle, the spin along $z$ and $x$ axes for Bob's particle should have both definite values (before Bob performs any spin measurement).

 

[PeterDonis]

The point is that, starting from Alice spin measurement on her particle, the spin along $z$ and $x$ axes for Bob's particle should have both definite values (before Bob performs any spin measurement).

The point is that, according to EPR's view of "reality", this should be the case, but QM says it isn't. As I said, EPR thought that meant QM had to be incomplete. Now, however, with Bell's theorem and experimental evidence that says the Bell inequalities are violated, we know it actually means that EPR's view of "reality" was wrong. Whatever "reality" is, it isn't what EPR thought it was.

 

[PeterDonis]

before Bob performs any spin measurement

According to QM, this doesn't matter. There is no quantum state at all that has definite values for two non-commuting observables. So Bob's particle can't have definite values for $z$ spin and $x$ spin at any time at all, whether before or after measurement, according to QM.

 

[cianfa72]

According to QM, this doesn't matter. There is no quantum state at all that has definite values for two non-commuting observables. So Bob's particle can't have definite values for $z$ spin and $x$ spin at any time at all, whether before or after measurement, according to QM.

Ok, since according to the math of QM there is not a common eigenvector/eigenstate for two non-commuting observable operators.

However I still don't have a crystal clear understanding of what EPR said. I believe EPR did not reject the uncertainty principle, so Alice cannot simultaneously measure both the spin of her particle along the $x$ and $z$ axes.

Therefore from the above statement does not follow that Bob's particle have definite values for both $z$ spin and $x$ spin.

 

[PeterDonis]

I believe EPR did not reject the uncertainty principle

Yes, they did, because they claimed that Bob's particle could have definite values for non-commuting observables. That is a rejection of the uncertainty principle.

 

[Nugatory]

so Alice cannot simultaneously measure both the spin of her particle along the x and z axes.

On the contrary, the EPR claim is that the spin on both axes can be measured by aligning one detector on the x axis and the other on the z axis. The crucial assumption, falsified by Bell tests, is that finding one particle spin-down on a given axis means that the other particle has the counterfactually definite property spin-up on that axis.

 

[DrChinese]

Ok, since according to the math of QM there is not a common eigenvector/eigenstate for two non-commuting observable operators.

However I still don't have a crystal clear understanding of what EPR said. I believe EPR did not reject the uncertainty principle, so Alice cannot simultaneously measure both the spin of her particle along the $x$ and $z$ axes.

Therefore from the above statement does not follow that Bob's particle have definite values for both $z$ spin and $x$ spin.

Agreeing 100% with everything that @PeterDonis is saying:

EPR said that *any* observable of Bob could be predicted with certainty by an appropriate measurement on Alice. They deducted (as have you) that Bob's observables must be predetermined (they did not use those words) and therefore QM was incomplete (because QM itself could not predict those values, and the canonical Uncertainty Principle seems to prevent it). Per QM, and in their words: "when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality." That is certainly a restatement of the uncertainty principle. So far, we should all agree.

But...to achieve their result, EPR had to make an explicit assumption, which seems reasonable on the face of it - but one that QM explicitly denies. They realized that only 1 observable could be predicted in advance. Since they also (implicitly) assumed that the choice of measurement on Alice should not affect distant Bob, they stated as follows (2nd to last paragraph of EPR, 1935):

"... Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does, not disturb the second system in any way. No reasonable definition of reality could be expected to permit this."

This is the point of departure between EPR and QM. EPR is assuming reality is objective, independent of the process of measurements made on a combined system. That's non-contextuality. The quantum mechanical expectation value for is explicitly contextual: the future measurement settings are the *only* parameters to the formula. So QM does not include their assumption.

 

[cianfa72]

"... On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does, not disturb the second system in any way. No reasonable definition of reality could be expected to permit this."

Ah ok, so the key point is that, in the last claim, EPR reject a definition of reality for quantities that depends upon the process of measurement carried out on the first system/particle. Therefore both quantities P and Q are "elements of reality" in contrast with uncertainty principle applied to a couple of non-commuting quantities/operators.

 

[cianfa72]

On the contrary, the EPR claim is that the spin on both axes can be measured by aligning one detector on the x axis and the other on the z axis.

You mean a detector aligned along the $x$ axis in Alice lab and another detector aligned along the $z$ axis in Bob lab (each of them measure just their own entangled particle).

 

[Nugatory]

You mean a detector aligned along the x axis in Alice lab and another detector along the z axis in Bob lab (each of them measure just their own particle).

The entire Alice/Bob two labs thing is just an aid to visualizing the important part, that there are two detectors. When I was doing this stuff back in college my two detectors were about 18 inches apart and I positioned them myself - conceptually no different than positioning them 18 feet apart and shouting the desired setting across the lab to a lab partner (who was named neither Alice nor Bob), or 18 miles apart and communicating it by telephone.

Likewise, many of the early Bell tests were done with two detectors pretty much side-by-side in the same lab and described as I did above. The physical separation and multiple lab setups are only needed when we're trying to close the locality loophole in a test of Bell's inequalities so want the setting event to be spacelike-separated from the emission event.

 

[cianfa72]

But...to achieve their result, EPR had to make an explicit assumption, which seems reasonable on the face of it - but one that QM explicitly denies. They realized that only 1 observable could be predicted in advance. Since they also (implicitly) assumed that the choice of measurement on Alice should not affect distant Bob, they stated as follows (2nd to last paragraph of EPR, 1935):

What is the precise EPR explict assumption that QM explicitly denies ?

This is the point of departure between EPR and QM. EPR is assuming reality is objective, independent of the process of measurements made on a combined system. That's non-contextuality. The quantum mechanical expectation value for is explicitly contextual: the future measurement settings are the *only* parameters to the formula. So QM does not include their assumption.

Sorry, could you be more specific about this point ? Thanks.

 

[DrChinese]

1. What is the precise EPR explict assumption that QM explicitly denies ?



Sorry, could you be more specific about this point [the future measurement settings are the *only* parameters to the formula] ? Thanks.

1. As quoted from EPR above:

The position of QM: "...since either one or the other, but not both simultaneously, of the [non-commuting] quantities P and Q can be predicted, they are not simultaneously real...This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does, not disturb the second system in any way."

Translation to modern terms: Reality is contextual; sometimes referred to as "non-realistic", "subjective" (based on chose of measurement basis" or "no hidden variables". Note that they slip in the assumption of locality in their statement of the QM position, by saying a measurement on one system does not disturb the other system in any way. This assumption can and has been challenged.

The position of EPR: "No reasonable definition of reality could be expected to permit this."

Comment: This is pure 100% assumption, and is not backed up in any way. Note that no one disputes the basic EPR idea that a measurement by Alice may be used to predict an outcome for Bob with certainty - as long as both measurement are made on the same basis.


2. The spin statistics for spin 1/2 particles is typically something like sin^2(Theta), and for spin 1 photons something like cos^2(Theta) - it varies a bit according to the precise setup. The common variable is then Theta. Theta is defined as the difference between the angle setting for Alice and the angle setting for Bob.

What is important about this is: a) the angle settings (and therefore Theta) are values in the future of the entangled pair, and can be set or changed midflight; and b) there are no other known variables or input parameters. If there were any such parameters to the predictive formula, then they all exactly cancel out - leaving us Theta alone.

So... this makes it difficult to point to hidden variables as being the source of the observed statistics if they all magically happen to cancel each other out. Hence: a future context (the measurement settings) is the only determinant of the observed statistics according to quantum theory, and no other factor has been discovered to date in experiments.

This is the very definition of contextuality. Quantum Mechanics is contextual. This is the opposite of what is usual for classical mechanics, in which one or more input variables at time T=0 participate in predicting the results at time T>0.

 

[cianfa72]

Following the QM standard math I tried to calculate the probability to get the opposite spin in a Bell experimental setup based on the measurement of the spin of a couple of entangled particles along three orthogonal axes $x$, $y$ and $z$.

We know that if Bob picks a different axis from Alice then QM claims there is a 50% probability to get spin-up or spin-down along his axis (regardless of the spin-up or spin-down result of Alice measurement on her axis).

Now assume Alice picks $z$ axis and measure spin-up along it. The probability to get spin-down for Bob spin measurement is

$$\left ( \frac {1} {3} + \frac {2} {3} \cdot \frac {1} {2} \right) = \frac {2} {3}$$
Since Alice and Bob pick at random the axis for spin measurement the Alice probability to pick for instance the $z$ axis and measure spin-up is 1/6, hence the total probability to get the opposite spin (either spin-up/spin-down or spin-down/spin-up) should be 2/3.

 

[PeterDonis]

We know that if Bob picks a different axis from Alice then QM claims there is a 50% probability to get spin-up or spin-down along his axis

No, that's not what we know.

We know that, if we just look at Bob's measurement and do not look at Alice's at all, then there is a 50% probability for Bob to get spin up or spin-down when measuring along any axis.

We know that, if we look at both measurements and we know Alice's result, then Bob's probability depends on the angle between his axis and Alice's axis.

We know that, if we look at both measurements and we don't know Alice's result, then Bob's probability is the same as for the first case above--Alice's measurement makes no difference to the probability we calculate for Bob unless we know Alice's result.

What I have just described, btw, is an example of the contextuality that @DrChinese is talking about. You can't make meaningful statements about probability in QM without specifying the context.

 

[cianfa72]

We know that, if we look at both measurements and we know Alice's result, then Bob's probability depends on the angle between his axis and Alice's axis.

Sorry, from Wikipedia if Alice picks $z$ axis and her result is spin-up $+z$ then the overall spin singlet system state $\ket {\psi}$ collapses in $$\ket {+z} \otimes \frac {\ket{+x} - \ket{-x}} {\sqrt 2}$$ hence Bob's measurement of $S_x$ returns either $+x$ or $-x$ with probability 1/2 each (in Bell setup we assume the angle between each pair of different axes $x$, $y$ or $z$ is always 90 degree). Same thing for Bob $S_y$ measurement.

Therefore the conditional probability for Bob's measurement on a different axis w.r.t. the knowledge of Alice's result along her axis is always 1/2.

We can use such probabilities to calculate the total probability to get opposite spin. Is my total probability calculation wrong ?

 

[Nugatory]

(in Bell setup we assume the angle between each pair of different axes x, y or z is always 90 degree)

In a bell test the measurement axes are not 90 degrees apart, because at that angle the inequality will not be violated - it’s the “equal” in the “less than or equal” formulation of the various forms of the inequality. Indeed, if we only consider tests with the axes at 90 degrees, there would be no experimental reason to reject the EPR argument that QM is incomplete.

 

[PeterDonis]

Sorry, from Wikipedia if Alice picks $z$ axis and her result is spin-up $+z$

Then you know Alice's result so you are in the second case I described.

then the overall spin singlet system state $\ket {\psi}$ collapses in $$\ket {+z} \otimes \frac {\ket{+x} - \ket{-x}} {\sqrt 2}$$ hence Bob's measurement of $S_x$ returns either $+x$ or $-x$ with probability 1/2 each

That's because the angle between Bob's and Alice's measurement axis is exactly 90 degrees. If the angle were different you would get different probabilities. As I said, if you know Alice's result, the probabilities you calculate for Bob depend on the angle between Bob's and Alice's measurement axes.

We can use such probabilities to calculate the total probability to get opposite spin.

What do you mean by this?

 

[cianfa72]

What do you mean by this?

I mean that, starting from those QM probabilities, one can calculate the total probability to get opposite spin in the Bell's experiment. If the 3 axes were mutually orthogonal then, according to my calculation in post#17, such a total probability should be 2/3.

In a Bell test the measurement axes are not 90 degrees apart, because at that angle the inequality will not be violated - it’s the “equal” in the “less than or equal” formulation of the various forms of the inequality. Indeed, if we only consider tests with the axes at 90 degrees, there would be no experimental reason to reject the EPR argument that QM is incomplete.

Ah ok, so if in Bell's test the three axes are for instance 120 degree apart, then the probability to get opposite spin along different axes (regardless of the specific pair of different axes that Alice and Bob pick) should be $$ \frac {1} {2} \left( 1 + cos \left( 2 \pi /3 \right ) \right) = \frac {1} {4} $$ hence the total probability to get opposite spin in the combined Bell's test is $$\left ( \frac {1} {3} + \frac {2} {3} \cdot \frac {1} {4} \right) = \frac {1} {2}$$

 

[PeterDonis]

I mean that, starting from those QM probabilities, one can calculate the total probability to get opposite spin in the Bell's experiment.

What does "the total probability" mean? Why is it even meaningful at all?

 

[cianfa72]

What does "the total probability" mean? Why is it even meaningful at all?

It is merely the probability for results of type "opposite spin" in a Bell's experimental setup -- i.e. it is the fraction of occurrences in which we get "opposite spin" in repeated Bell's tests.

 

[PeterDonis]

It is merely the probability for results of type "opposite spin" in a Bell's experimental setup -- i.e. it is the fraction of occurrences in which we get "opposite spin" in repeated Bell's tests.

There is no such thing unless you define what measurements are being made. And the result can be different for different specifications of the measurements. There is no single "total probability". So what you are trying to calculate is meaningless.

 

[cianfa72]

There is no single "total probability". So what you are trying to calculate is meaningless.

Maybe I'm wrong but in a Bell's experimental setup (in which Alice and Bob measure her/his particle spin along 3 axes 120 degree apart) the EPR predicted probability for "opposite spin" is at least 5/9, whereas the standard QM calculation prediction is just 1/2 (see my post#22).

 

[Nugatory]

Maybe I'm wrong but in a Bell's experimental setup (in which Alice and Bob measure her/his particle spin along 3 axes 120 degree apart) the EPR predicted probability for "opposite spin" is at least 5/9, whereas the standard QM calculation prediction is just 1/2 (see my post#22).

I am not sure what you mean by “the EPR predicted probability”, but everyone agrees that the probability of opposite spins is a function of the angle between the two axes, and ranges from 100% when that angle is zero to 50% when that angle is 90 degrees and zero when that angle is 180 degrees.

Difficulties only appear if we accept the EPR assumption (this is what you started the thread about, right?) that measuring one particle spin-up on a given axis implies that the other particle has the property spin-down on that axis. That assumption leads to Bell’s inequality, which has been observed to be violated.

 

[PeterDonis]

in a Bell's experimental setup

In other words, once you specify which measurements are being made. Yes, but the probability you calculate is only valid for that specific setup. You appeared to be claming that you could calculate some sort of "total probability" that applied to any experimental setup. You can't.

 

[cianfa72]

In other words, once you specify which measurements are being made. Yes, but the probability you calculate is only valid for that specific setup. You appeared to be claming that you could calculate some sort of "total probability" that applied to any experimental setup.

No no...that's not what I meant. Just to be clear I was referring to the specific setup in which Alice and Bob measure their own particle's spin of an entangled pair randomly picking an axis from a set of 3 axes 120 degree apart.

Using standard QM the probability for "opposite spin" along different picked axes (always 120 degree apart) is 1/4. The probability that Alice and Bob randomly pick two different axes is 2/3 while the probability to randomly pick the same axis is 1/3 (in this case the QM probability to get "opposite spin" is 1). Hence, using the rules of probability calculation, we get $$\left ( \frac {2} {3} \cdot \frac {1} {4} + \frac {1} {3} \right) = \frac {1} {2}$$ for the probability to get "opposite spin" in this specific experimental setup.

 

[PeterDonis]

I was referring to the specific setup in which Alice and Bob measure their own particle's spin of an entangled pair randomly picking an axis from a set of 3 axes 120 degree apart.

Ok. But note that that is not the same as what is done to test the Bell inequalities, or other related ones such as CHSH. To do such a test you need to make measurements of all of the possible combinations and then do the appropriate calculations from the results.