Why the Quantum | A Response to Wheeler's 1986 Paper - Comments

[RUTA]

Greg Bernhardt submitted a new PF Insights post

Why the Quantum | A Response to Wheeler's 1986 Paper

Continue reading the Original PF Insights Post.

 

[stevendaryl]

So the quantum weirdness in an EPR-type experiment is due to a combination of two things, neither of which is weird in itself:

1. Conservation laws (conservation of angular momentum)
2. Discreteness of measurement results (always getting $\pm \frac{\hbar}{2}$ for the spin measurement in any direction)

But it seems that there is something else going on in EPR, which is a collapse-like assumption: When you measure a fermion's spin along some axis $\vec{a}$, then it is as if, afterward, it is definitely in that direction. That's different from an imagined classical measurement that is somehow constrained to give a discrete result. You could imagine (this is Bell's toy model) that the electron as an associated spin vector, $\vec{s}$, and measuring spin with respect to an axis $\vec{a}$ would return $+1/2$ if the angle between $\vec{s}$ and $\vec{a}$ is less than 90^o, and $-1/2$ otherwise. This would give a discrete result, but the result would not be the actual spin vector of the electron.

 

[RUTA]

So the quantum weirdness in an EPR-type experiment is due to a combination of two things, neither of which is weird in itself:

1. Conservation laws (conservation of angular momentum)
2. Discreteness of measurement results (always getting $\pm \frac{\hbar}{2}$ for the spin measurement in any direction)

But it seems that there is something else going on in EPR, which is a collapse-like assumption: When you measure a fermion's spin along some axis $\vec{a}$, then it is as if, afterward, it is definitely in that direction. That's different from an imagined classical measurement that is somehow constrained to give a discrete result. You could imagine (this is Bell's toy model) that the electron as an associated spin vector, $\vec{s}$, and measuring spin with respect to an axis $\vec{a}$ would return $+1/2$ if the angle between $\vec{s}$ and $\vec{a}$ is less than 90^o, and $-1/2$ otherwise. This would give a discrete result, but the result would not be the actual spin vector of the electron.

Thnx for your comments, stevendaryl. 1 and 2 are spot on, but the collapse of some definite vector in that fashion doesn’t reproduce the quantum correlations (see the example in the Dehlinger paper referenced therein). The quantum correlations assume +1 or -1 is the “magnitude” and either Alice or Bob can claim they are always measuring the magnitude in each trial, it’s the other person who is getting the average, no collapse necessary. How can you not be impressed with such perspectival invariance?

 

[bhobba]

Very nice post/paper

My thoughts on the matter have tended to be more along the lines of generalized probability models and QM being the simplest one after ordinary probability theory that allows continuous transformations between pure states. But it is quite abstract - yours is much more physical.

Very thought provoking.

Thanks
Bill

 

[stevendaryl]

I'm trying to get an intuitive understanding of the way that EPR probabilities (for anti-correlated spin-1/2 particles) are in some sense the closest we can get to the criteria:

1. The sum of the spins is zero.
2. Spin measurement always gives $\pm 1/2$

If Alice measures her particle's spin along axis $A$ and Bob measures his particle's spin along axis $B$, then it is impossible to satisfy both criteria, because unless $A$ and $B$ are aligned, none of the following combinations adds up to zero:

1. $\frac{1}{2} (+\vec{A} + \vec{B})$
2. $\frac{1}{2} (+\vec{A} - \vec{B})$
   
3. $\frac{1}{2} (-\vec{A} + \vec{B})$
   
4. $\frac{1}{2} (-\vec{A} - \vec{B})$

What the quantum probabilities do instead is the following:

• Filter only those events in which Alice gets +1/2. (That includes possibilities 1&2 above)
• Compute the vectorial average of the spin sums: This will be given by: $\frac{1}{2} (P_1 (\vec{A} + \vec{B}) + P_2 (\vec{A} - \vec{B}))$ (where $P_1$ is the probability of possibility 1 above, and $P_2$ is the probability of possibility 2).
• This average is still not zero, but its projection onto $\vec{B}$ is zero.

This uniquely determines the probabilities $P_1$ and $P_2$:

1. $P_1 + P_2 = 1$
2. $(\frac{1}{2} (P_1 (\vec{A} + \vec{B}) + P_2 (\vec{A} - \vec{B}))) \cdot \vec{B} = 0$

The latter equation becomes:

$\frac{1}{2} (P_1 (cos(\theta) + 1) + P_2 (cos(\theta) - 1)) = 0$ (where $\theta$ is the angle between $A$ and $B$)

These equations have the unique solution: $P_1 = \frac{1}{2} (1-cos(\theta)) = sin^2(\frac{\theta}{2})$, $P_2 = \frac{1}{2} (1+cos(\theta)) = cos^2(\frac{\theta}{2})$

Those are the quantum probabilities for anti-correlated spin-1/2 particles.

That's sort of interesting, but my understanding of the motivation is a little muddled. I understand that you can't have perfect cancellation if the axes $\vec{A}$ and $\vec{B}$ are not aligned. But why ask for cancellation (on the average) along axis $\vec{B}$?

 

[RUTA]

I'm trying to get an intuitive understanding of the way that EPR probabilities (for anti-correlated spin-1/2 particles) are in some sense the closest we can get to the criteria:

1. The sum of the spins is zero.
2. Spin measurement always gives $\pm 1/2$

... Those are the quantum probabilities for anti-correlated spin-1/2 particles.

That's sort of interesting, but my understanding of the motivation is a little muddled. I understand that you can't have perfect cancellation if the axes $\vec{A}$ and $\vec{B}$ are not aligned. But why ask for cancellation (on the average) along axis $\vec{B}$?

You've arrived at the heart of the spin singlet state (uniquely producing the max deviation from the CHSH-Bell inequality, i.e., the Tsirelson bound). There is nothing special about axis $\vec{B}$, indeed you could have done the analysis looking at Bob's results along axis $\vec{B}$ and required projection along axis $\vec{A}$ to be zero in producing the correlation $\langle a,b \rangle$ . So, in which direction is angular momentum for the quantum exchange of momentum actually being conserved? It's being conserved on average from either Bob or Alice's perspective, i.e., along either $\vec{B}$ or $\vec{A}$. In classical physics there is a definite direction for angular momentum $\vec{L}$, so with Alice and Bob measuring along random directions in the classical case we would expect neither $\vec{A}$ nor $\vec{B}$ to align with $\vec{L}$. Consequently, in the classical case, Alice and Bob should always (essentially) be measuring something less than the magnitude L of the conserved quantity $\vec{L}$ (as shown in the picture of the SG experiment in the Insight). But, in the quantum case, it's as if there is no $\vec{L}$ independent of Alice and Bob's measurements. That is, you can't account for the quantum correlation using a hidden variable or Mermin "instruction sets" on a trial-by-trial basis (giving classical correlations satisfying the Bell inequality). No, the bottom line is that the quantum correlation satisfies a truly frame-independent conservation principle.

As I said in the Insight, this is reminiscent of another frame-independent principle, the light postulate of SR. That postulate also led to "weird consequences," e.g., length contraction, time dilation, and relativity of simultaneity, and it was also opposed because it was something postulated not explained. Making this reference to the light postulate was motivated by quotes from Hardy and other reconstructionists in QIT. You read in many places in the QIT literature things like this Hardy quote

The standard axioms of QT [quantum theory] are rather ad hoc. Where does this structure come from? Can we write down natural axioms, principles, laws, or postulates from which [we] can derive this structure? Compare with the Lorentz transformations and Einstein's two postulates for special relativity. Or compare with Kepler's Laws and Newton's Laws. The standard axioms of quantum theory look rather ad hoc like the Lorentz transformations or Kepler's laws. Can we find a natural set of postulates for quantum theory that are akin to Einstein's or Newton's laws?

So, in our paper we point out that this explanation of the Tsirelson bound should satisfy the desideratum of QIT.

We're still waiting for Bub's response, he was the one who asked us to bring our adynamical approach to bear on his question "Why the Tsirelson bound?" when we gave a talk on our book at his QIT seminar last April. He wrote a nice blurb for that book, so we're hoping he now sees the relevance of adynamical/constraint-based explanation for QIT.

Any suggestions for where to submit the paper? I was thinking PRA, since they do QIT stuff.

 

[RUTA]

What the quantum probabilities do instead is the following:

• Filter only those events in which Alice gets +1/2. (That includes possibilities 1&2 above)
• Compute the vectorial average of the spin sums: This will be given by: $\frac{1}{2} (P_1 (\vec{A} + \vec{B}) + P_2 (\vec{A} - \vec{B}))$ (where $P_1$ is the probability of possibility 1 above, and $P_2$ is the probability of possibility 2).
• This average is still not zero, but its projection onto $\vec{B}$ is zero.

I understand that you can't have perfect cancellation if the axes $\vec{A}$ and $\vec{B}$ are not aligned. But why ask for cancellation (on the average) along axis $\vec{B}$?

To fill in the blanks (not for you, I know you get it, but for others who might not have followed our exchange), rewrite $\frac{1}{2} (P_1 (\vec{A} + \vec{B}) + P_2 (\vec{A} - \vec{B}))\cdot\vec{B}$ as $\frac{1}{2} ((P_1 + P_2)\vec{A} + (P_1 - P_2)\vec{B}))\cdot\vec{B} = \frac{1}{2} (\vec{A} + (P_1 - P_2)\vec{B}))\cdot\vec{B}$ where $\frac{1}{2}$ is the magnitude of Alice's measurement along $\vec{A}$ (note that both $\vec{A}$ and $\vec{B}$ are unit vectors). Now we're only considering those outcomes for which Alice measured $+\frac{1}{2}$ (first bullet point), so the average value Alice would expect to measure along $\vec{B}$ for her $+\frac{1}{2}$ outcomes along $\vec{A}$ is $+\frac{1}{2}\vec{A}\cdot\vec{B} = +\frac{1}{2}cos(\theta)$. Since we need $\vec{L}$ conserved to zero on average, we need Bob's average result along $\vec{B}$ to cancel this $+\frac{1}{2}cos(\theta)$. His average is $+\frac{1}{2}P_1 + -\frac{1}{2}P_2 = (+\frac{1}{2}P_1\vec{B} + -\frac{1}{2}P_2\vec{B})\cdot\vec{B}$. Thus, we need $+\frac{1}{2}cos(\theta) + \frac{1}{2}P_1 + -\frac{1}{2}P_2 = \frac{1}{2} (\vec{A} + (P_1 - P_2)\vec{B}))\cdot\vec{B} = 0$. Again, you can divide up the results the same way for Bob and demand that Alice's average outcomes cancel his $+\frac{1}{2}cos(\theta)$ to derive the same quantum correlations.

 

[bhobba]

he was the one who asked us to bring our adynamical approach to bear on his question "Why the Tsirelson bound?" when we gave a talk on our book at his QIT seminar last April.

Have ordered a copy from Amazon.

Looking forward to reading it.

Interesting Hardy is the one that got me into the probabilistic view of QM foundations.

Do you know if he has moved away from that?

Thanks
Bill

 

[RUTA]

Interesting Hardy is the one that got me into the probabilistic view of QM foundations.

Do you know if he has moved away from that?

Thanks
Bill

Hardy revised his original (2001) set of axioms "replacing the simplicity axiom with more a compelling axiom" in 2011 (https://arxiv.org/abs/1104.2066). Per Hardy, "We show that classical probability theory and quantum theory are the only two theories consistent with the following set of postulates." His new postulates are Sharpness, Information Locality, Tomographic Locality, Permutability, and Sturdiness, which follow from two simple axioms:

Axiom 1 Operations correspond to operators.
Axiom 2 Every complete set of physical operators corresponds to a complete set of operations.

In the original version of our paper (as presented in the IJQF workshop last month), we advocated "quantum-classical contextuality," where physical reality isn't "quantum rather than classical, but fundamentally both." Thus, we made explicit reference to Hardy's 2011 statement and postulates. We nixed that when we decided to submit the paper to a physics journal.

Be forewarned about our book -- as a mathematician, you'll want to avoid the philosophical threads. The main thread is probably already too philosophical for you :-)

 

[vanhees71]

I'm trying to get an intuitive understanding of the way that EPR probabilities (for anti-correlated spin-1/2 particles) are in some sense the closest we can get to the criteria:

1. The sum of the spins is zero.
2. Spin measurement always gives $\pm 1/2$

If Alice measures her particle's spin along axis $A$ and Bob measures his particle's spin along axis $B$, then it is impossible to satisfy both criteria, because unless $A$ and $B$ are aligned, none of the following combinations adds up to zero:

1. $\frac{1}{2} (+\vec{A} + \vec{B})$
2. $\frac{1}{2} (+\vec{A} - \vec{B})$
   
3. $\frac{1}{2} (-\vec{A} + \vec{B})$
   
4. $\frac{1}{2} (-\vec{A} - \vec{B})$

What the quantum probabilities do instead is the following:

• Filter only those events in which Alice gets +1/2. (That includes possibilities 1&2 above)
• Compute the vectorial average of the spin sums: This will be given by: $\frac{1}{2} (P_1 (\vec{A} + \vec{B}) + P_2 (\vec{A} - \vec{B}))$ (where $P_1$ is the probability of possibility 1 above, and $P_2$ is the probability of possibility 2).
• This average is still not zero, but its projection onto $\vec{B}$ is zero.

This uniquely determines the probabilities $P_1$ and $P_2$:

1. $P_1 + P_2 = 1$
2. $(\frac{1}{2} (P_1 (\vec{A} + \vec{B}) + P_2 (\vec{A} - \vec{B}))) \cdot \vec{B} = 0$

The latter equation becomes:

$\frac{1}{2} (P_1 (cos(\theta) + 1) + P_2 (cos(\theta) - 1)) = 0$ (where $\theta$ is the angle between $A$ and $B$)

These equations have the unique solution: $P_1 = \frac{1}{2} (1-cos(\theta)) = sin^2(\frac{\theta}{2})$, $P_2 = \frac{1}{2} (1+cos(\theta)) = cos^2(\frac{\theta}{2})$

Those are the quantum probabilities for anti-correlated spin-1/2 particles.

That's sort of interesting, but my understanding of the motivation is a little muddled. I understand that you can't have perfect cancellation if the axes $\vec{A}$ and $\vec{B}$ are not aligned. But why ask for cancellation (on the average) along axis $\vec{B}$?

I'm not sure, whether I understand your problem. This is an example for the fact that a single-particle quantity (like single-particle spin) in a many-body system can be determined for the system as a whole (here the total angular momentum) while the single-particle quantities are indetermined. That's an implication of entanglement.

In the here discussed case you have a total angular momentum 0 state of a two-particle system of spin-1/2 particles, i.e., the two-particle spin state is
$$|\Psi \rangle=\frac{1}{\sqrt{2}} (|1/2,-1/2 \rangle - |-1/2,1/2 \rangle).$$
That's obviously a simultaneous eigenstate of $|\hat{\vec{S}}^2 \rangle$ and $|\hat{S}_z \rangle$ to the eigenvalues $S=0$, $\sigma_z=0$. Here
$$\hat{\vec{S}}=\hat{\vec{s}} \otimes \hat{1} + \hat{1} \otimes \hat{\vec{s}}.$$
Note that the $S=0$ state is very special, because in this case all three components of $\vec{S}$ are determined although these operators do not commute.

Nevertheless the single-particle spins are maximally undetermined, i.e., there probabilities are given by the Statistical operator
$$\hat{\rho}_A=\mathrm{Tr}_B |\Psi \rangle \langle \Psi|=\frac{1}{2} \hat{1}$$
and
$$\hat{\rho}_B=\mathrm{Tr}_A |\Psi \rangle \langle \Psi|=\frac{1}{2} \hat{1}.$$
Now measuring the angular momentum component at particles A and B in different directions you get the probabilities you quote, and that's all you know about the outcome of measurements of the single-particle angular momenta. Of course, the measured outcomes do not add up to 0. Why should they? Even in classical physics it doesn't make too much sense to add components of vectors with respect to basis vectors in different directions. Of course, if you measure the angular-momentum components for both particles wrt. the same direction, then they must add up to 0 due to the preparation of the two-body system in the $S=0$ state. As explained above here you have the special case of a preparation of all three angular-momentum components to have the determined value 0. This is special, because that's possible only for the $S=0$ state and is due to the complete rotational symmetry (isotrophy) of this one special state. So you can have sometimes common eigenvectors for incompatible observables, and that's the most common example for this fact.

I don't see any further specialty in this example, despite the fact that it's the most simple example to explain entanglement, Bell's inequality and all such unusual quantum correlations without a classical counterpart. It's only a problem, if you don't accept the quite abstract mathematical formulation of quantum theory and its minimal probabilistic interpretation in terms of Born's rule. Due to our persistent intuition from our experience with classically behaving (quantum) objects (aka many-many-many-...-body systems) we sometimes think we have to "explain" something more with quantum theory than there is contained in it, but that's pretty misleading.

 

[stevendaryl]

I'm not sure, whether I understand your problem. This is an example for the fact that a single-particle quantity (like single-particle spin) in a many-body system can be determined for the system as a whole (here the total angular momentum) while the single-particle quantities are indetermined. That's an implication of entanglement.

In the here discussed case you have a total angular momentum 0 state of a two-particle system of spin-1/2 particles, i.e., the two-particle spin state is
$$|\Psi \rangle=\frac{1}{\sqrt{2}} (|1/2,-1/2 \rangle - |-1/2,1/2 \rangle).$$

The issue was not to derive the quantum probabilities from quantum mechanics, but to see if those probabilities can be derived from the assumptions that:

1. The measured angular momenta of the two particles separately yields a discrete answer: $\pm \frac{\vec{A}}{2}$ for the first measurement and $\pm \frac{\vec{B}}{2}$ for the second measurement.
2. The sum of the spins must add up to zero (in some average sense).

 

[bhobba]

Be forewarned about our book -- as a mathematician, you'll want to avoid the philosophical threads. The main thread is probably already too philosophical for you :-)

Don't worry - I sort of figured that out with its emphasis on the Blockworld which I am not a fan of. But as I often say - my views mean jack shite - I am sure it will contain interesting insights. Every interpretation of QM I have read, transnational, MW, BM etc etc have helped me in understanding the formalism better. My personal views are pretty well known, and are virtually identical to Vanhees, but to hold any view you must subject it to what other views say.

Thanks
Bill

 

[bhobba]

The issue was not to derive the quantum probabilities from quantum mechanics, but to see if those probabilities can be derived from the assumptions that:

1. The measured angular momenta of the two particles separately yields a discrete answer: $\pm \frac{\vec{A}}{2}$ for the first measurement and $\pm \frac{\vec{B}}{2}$ for the second measurement.
2. The sum of the spins must add up to zero (in some average sense).

This whole approach is very new to me. I am very interested in QM foundations, but have been taking a back seat and listening, rather than participating until I feel more comfortable commenting.

Thanks
Bill

 

[vanhees71]

The issue was not to derive the quantum probabilities from quantum mechanics, but to see if those probabilities can be derived from the assumptions that:

1. The measured angular momenta of the two particles separately yields a discrete answer: $\pm \frac{\vec{A}}{2}$ for the first measurement and $\pm \frac{\vec{B}}{2}$ for the second measurement.
2. The sum of the spins must add up to zero (in some average sense).

That's a different question, which is not describing the statistics you expect from quantum theory as detailed above!

 

[vanhees71]

Be forewarned about our book -- as a mathematician, you'll want to avoid the philosophical threads. The main thread is probably already too philosophical for you :-)

Thanks for the warning. In general, philosophy on quantum theory rather confuses the reader than to help him or her. Maybe your book is an exception. Nevertheless the word philosophy in connection with quantum theory (or even physics in general) should be read as a caveat sign ;-)).

 

[stevendaryl]

That's a different question, which is not describing the statistics you expect from quantum theory as detailed above!

I'm not sure what you mean. Both the heuristic argument (not original with me; I was paraphrasing the Insights article) and the quantum theory make the same predictions: If Alice measures her particle's spin along axis $\vec{A}$ and Bob measures his particle's spin along axis $\vec{B}$, then the conditional probabilities are:

• $P_1$ = the probability that Bob will measure spin-up given that Alice measures spin-up = $sin^2(\frac{\theta}{2})$
• $P_2$ = the probability that Bob will measure spin-down given that Alice measures spin-up = $cos^2(\frac{\theta}{2})$

(where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$).

 

[vanhees71]

Yes, but the spin components in non-collinear directions need not to cancel each other. Why should they?

 

[stevendaryl]

Yes, but the spin components in non-collinear directions need not to cancel each other. Why should they?

That was what I asking RUTA for clarification about.

The sense in which there is cancellation on the average is this:

Among those events where Alice measures spin-up along direction $\vec{A}$, the expectation for
$(\vec{S}_A + \vec{S}_B) \cdot \vec{B}$ = 0

(where $\vec{S}_A = + \frac{1}{2} \vec{A}$ and $\vec{S}_B = \pm \frac{1}{2} \vec{B}$, depending on whether Bob gets spin-up or spin-down)

 

[vanhees71]

It's again not clear to me what you want to calculate. The joint probability that A finds $\sigma_A=\pm 1/2$ and B find $\sigma_B=\pm 1/2$ is, of course
$$P(\sigma_A,\sigma_B) =|\langle \sigma_A | \otimes \langle \sigma_B |\Psi \rangle|^2.$$
Here $|\sigma_A \rangle$ and $|\sigma_B \rangle$ are the eigenvalues of the operators
$$\hat{\sigma}_{A}=\vec{A} \cdot \hat{\vec{\sigma}}, \quad \hat{\sigma}_{A}=\vec{A} \cdot \hat{\vec{\sigma}}.$$
I'm to lazy to explicitly figure this out, but I still don't see the point of the exercise :-(.

 

[RUTA]

Thanks for the warning. In general, philosophy on quantum theory rather confuses the reader than to help him or her. Maybe your book is an exception. Nevertheless the word philosophy in connection with quantum theory (or even physics in general) should be read as a caveat sign ;-)).

There are philosophers who are interested in foundations of physics (FoP) and our book was written for them as well as physicists interested in FoP. As a typical physicist, I tend to make unarticulated assumptions and the philosophers are good at identifying those. My interest in FoP is based on my desire for a model of objective reality for all of physics. See Becker's book for the value in this.

 

[stevendaryl]

It's again not clear to me what you want to calculate.

I thought I was pretty clear, but I can try again. Forget quantum mechanics for a second, and let's consider the following purely mathematical problem:

1. There is a sequence of pairs of particles.
2. Out of each pair, Alice measures the spin $\vec{S}_A$ of one of the particles.
3. Bob measures the spin $\vec{S}_B$ of the other particle.
4. For whatever reason, Alice always gets the answer $\vec{S}_A = \pm \frac{1}{2} \vec{A}$
5. Bob always gets the answer $\vec{S}_B = \pm \frac{1}{2} \vec{B}$
6. Let $P_1$ be the conditional probability that Bob gets $+\frac{1}{2} \vec{B}$ given that Alice measures $+\frac{1}{2} \vec{A}$
7. Let $P_2$ be the conditional probability that Bob gets $-\frac{1}{2} \vec{B}$ given that Alice measures $+\frac{1}{2} \vec{A}$
   
8. Assume that out of those events where Alice gets $\vec{S}_A = +\frac{1}{2} \vec{A}$, the expected value of $(\vec{S}_A + \vec{S}_B) \cdot \vec{B}$ is zero.

Question: Find $P_1$ and $P_2$

So it's a purely mathematical problem. The claim being made is that 1-8 allows you to deduce the answer to the question. You cannot bring up quantum mechanics to answer the question, because that's not one of the assumptions 1-8.

But the connection with quantum mechanics is that the conditional probabilities--$P_1$ = the conditional probability that Bob measures spin-up along $\vec{B}$ given that Alice measures spin-up along $\vec{A}$, and $P_2$ = the probability that Bob measures spin-down--are the same as the quantum prediction for EPR.

I don't understand what you find confusing. Computing the quantum prediction of relative probabilities, or the statement of the problem?

 

[stevendaryl]

I don't understand what you find confusing. Computing the quantum prediction of relative probabilities, or the statement of the problem?

The third possibility, which I also find confusing, is exactly what the significance of the argument is.

 

[RUTA]

I thought I was pretty clear, but I can try again. Forget quantum mechanics for a second, and let's consider the following purely mathematical problem:

1. There is a sequence of pairs of particles.
2. Out of each pair, Alice measures the spin $\vec{S}_A$ of one of the particles.
3. Bob measures the spin $\vec{S}_B$ of the other particle.
4. For whatever reason, Alice always gets the answer $\vec{S}_A = \pm \frac{1}{2} \vec{A}$
5. Bob always gets the answer $\vec{S}_B = \pm \frac{1}{2} \vec{B}$
6. Let $P_1$ be the conditional probability that Bob gets $+\frac{1}{2} \vec{B}$ given that Alice measures $+\frac{1}{2} \vec{A}$
7. Let $P_2$ be the conditional probability that Bob gets $-\frac{1}{2} \vec{B}$ given that Alice measures $+\frac{1}{2} \vec{A}$
   
8. Assume that the expected value of $(\vec{S}_A + \vec{S}_B) \cdot \vec{B}$ is zero.

Question: Find $P_1$ and $P_2$

So it's a purely mathematical problem. The claim being made is that 1-8 allows you to deduce the answer to the question. You cannot bring up quantum mechanics to answer the question, because that's not one of the assumptions 1-8.

But the connection with quantum mechanics is that the conditional probabilities--$P_1$ = the conditional probability that Bob measures spin-up along $\vec{B}$ given that Alice measures spin-up along $\vec{A}$, and $P_2$ = the probability that Bob measures spin-down--are the same as the quantum prediction for EPR.

I don't understand what you find confusing. Computing the quantum prediction of relative probabilities, or the statement of the problem?

One suggested addition—items 6-8 are from Alice’s perspective. Changing to Bob’s perspective you would be dotting along $\vec{A}$ in 8. Either way gives the QM result.

 

[RUTA]

Let me also point out that these calculations are just an example of what we already know about QM. We know entangled states are created from conservation principles and that QM states reproduce CM via expectation values, i.e., on average. So, of course, this result has to follow. I thought it was interesting to see exactly what the general idea (QM gives CM on average) means for these states, but not the least bit surprising. That was Unnikrishnan's attitude in the paper referenced in my Insight -- he asked (polemically) why anyone would even bother to check for violations of the Bell inequality. Why would anyone expect classical probability to hold when classical probability would violate conservation of L? Given there are quantum exchanges of momentum, classical probability theory cannot possibly provide for conservation of L, it just cannot hold on a trial-by-trial basis for quantum exchange of momentum.

So, that the Tsirelson bound (extent to which the Bell inequality is violated by QM) follows from the conservation of L for the quantum exchange of momentum is merely showing us an implication of QM --> CM on average when the fundamental exchanges of momentum are quantized. Again, I thought it was cool to see exactly what that means for the spin singlet state per Unnikrishnan and then figuring out what it means for the Mermin photon state myself. These examples really clarified the relationship between QM and CM for me by providing a more physical basis for what I had already written in our book.

And, that relationship provides a beautifully self-consistent model of objective reality (without instrumentalism) as long as you don't require a dynamical model. This example does absolutely nothing to help those stuck in the "ant's-eye view." That's the point of our book and that's the point of this Insight (which is why it's linked to my BW series).

The other thing I learned from these examples is the apparent importance of no preferred reference frame in Nature. The relativity principle, the light postulate, and now the direction-invariant manner by which QM gives rise to conservation of L all speak to the fundamental importance of no preferred reference frame. I've been studying physics for almost 40 years and I'm still discovering elements of its beauty :-)

 

[vanhees71]

The other thing I learned from these examples is the apparent importance of no preferred reference frame in Nature. The relativity principle, the light postulate, and now the direction-invariant manner by which QM gives rise to conservation of L all speak to the fundamental importance of no preferred reference frame. I've been studying physics for almost 40 years and I'm still discovering elements of its beauty :-)

Perhaps I should read more carefully your insights article, but this line of arguments is very strange to me. In classical Newtonian as well as special relativistic physics the total angular momentum of a closed system is conserved by construction since it follows from the isotropy of both Galilei-Newton as well we Einstein-Minkowski spacetime. In classical statistical mechanics this still holds strictly true too. So I don't get the point of this argument.

There's also no preferred reference frame in both classical and quantum theory by construction. Again it's a mathematical consequenz of Galilei or Poincare invariance of the physical laws. Indeed, the geometrical approach in a modern sense is a great element of beauty, and I don't see any necessity to destroy this beauty!

 

[vanhees71]

But the connection with quantum mechanics is that the conditional probabilities--$P_1$ = the conditional probability that Bob measures spin-up along $\vec{B}$ given that Alice measures spin-up along $\vec{A}$, and $P_2$ = the probability that Bob measures spin-down--are the same as the quantum prediction for EPR.

I don't understand what you find confusing. Computing the quantum prediction of relative probabilities, or the statement of the problem?

Ok, then I misunderstood the purpose of this entire discussion. I thought it was about quantum mechanics. I don't see any clear classical-statistical physics picture of the quite complicated probability-theory exercise either. Still puzzled...

 

[RUTA]

Perhaps I should read more carefully your insights article, but this line of arguments is very strange to me. In classical Newtonian as well as special relativistic physics the total angular momentum of a closed system is conserved by construction since it follows from the isotropy of both Galilei-Newton as well we Einstein-Minkowski spacetime. In classical statistical mechanics this still holds strictly true too. So I don't get the point of this argument.

There's also no preferred reference frame in both classical and quantum theory by construction. Again it's a mathematical consequenz of Galilei or Poincare invariance of the physical laws. Indeed, the geometrical approach in a modern sense is a great element of beauty, and I don't see any necessity to destroy this beauty!

But, in quantum mechanics, we can have conservation of a directional quantity with no preferred direction! That doesn't even make sense classically where you're only going to measure a fraction of the magnitude of the conserved vector quantity when you measure in another direction. That's pretty cool.

 

[vanhees71]

What do you mean by this?

In classical theory you also have conservation of angular momentum for any closed system. If you take some bomb sitting somewhere at rest, and it's exploding without any outside influence (e.g., by some time fuse within the bomb itself triggering the explosion, i.e., without any external transfer of angular momentum) the total angular momentum of the pieces flying apart is still 0. This is qualitatively not different from the quantum-mechanical example of a decaying (pseudo-)scalar particle into two spin-1/2 particles (e.g., $\pi^+ \rightarrow \mu^+ + \nu_{\mu}$). The total angular momentum in the rest frame of the pion (center-mass frame of the muon and muon-neutrino) is 0.

I don't understand the statement about the measurement on a conserved vector quantity.

 

[stevendaryl]

Ok, then I misunderstood the purpose of this entire discussion. I thought it was about quantum mechanics.

Well, it is in the sense that the exercise leads to the same conditional probabilities as QM.

 

[RUTA]

What do you mean by this?

In classical theory you also have conservation of angular momentum for any closed system. If you take some bomb sitting somewhere at rest, and it's exploding without any outside influence (e.g., by some time fuse within the bomb itself triggering the explosion, i.e., without any external transfer of angular momentum) the total angular momentum of the pieces flying apart is still 0. This is qualitatively not different from the quantum-mechanical example of a decaying (pseudo-)scalar particle into two spin-1/2 particles (e.g., $\pi^+ \rightarrow \mu^+ + \nu_{\mu}$). The total angular momentum in the rest frame of the pion (center-mass frame of the muon and muon-neutrino) is 0.

I don't understand the statement about the measurement on a conserved vector quantity.

For the bomb, you add up the momenta of all the pieces and get zero. For the decay of a neutral pi meson, the electron and positron will only give zero total spin if you measure each piece along the same axis.

 

[RUTA]

I said something that may have contributed to vanhees71’s confusion. I said these states have no preferred direction for a conserved vector quantity. Well since the conserved vector is null that’s a stupid thing to say. I should have said Alice and Bob are always measuring non-zero L that always cancel when co-aligned. So when not co-aligned we expect fractional results from either at minimum. Instead these two vectors are always the same length such that either cancels the other on average. That’s the sense in which we have conservation of a vector quantity with no preferred direction.

Edit: See my detailed explanation in #33 below.

 

[vanhees71]

Yes, but you'll get zero by measuring the angular momentum in any direction. I guess, I now get the problem you want to discuss. Of course, you can for each single decay only measure the angular momentum in one direction, not in three linearly independent ones since angular-momentum components at different directions are not compatible to each other.

Now all the quibbles with this gets resolved, when you take the minimal statistical interpretation seriously and accept that the meaning of quantum states are probabilities for the outcome of measurements according to Born's rule and nothing else (and you cannot know more, if QT is correct, which I assume due to the lack of any contradictions of experience to the predictions of QT): To verify the probabilistic predictions of quantum theory you have to consider an ensemble of very many decaying particles and measure the angular-momentum components in three linearly independent on a sufficiently large subensemble for each direction since you can only measure one component for each single event. The prediction of QT is a 1:1 correlation between the outcomes of A's and B's measurement of the spin components of the decay particles in the same direction, and this holds true for any direction, and this is in full accordance with angular-momentum conservation. Of course the outcome of these measurements is completely random, but the correlation holds strictly true (with 100% probability).

As in all cases of apparent "quantum weirdness" I know, the minimal statistical interpretation resolves the weirdness. The only weirdness remaining is due to our classically trained prejudices about the behavior of objects, but these prejudices are due to our everyday experience with very much coarse-grained macroscopic observables, which are in fact averaging over many microscopic degrees of freedom, which leads to an apparent classical behavior, but in fact it's just due to the sufficiency of coarse-grained macroscopic observables to describe macroscopic systems. On these macroscopic scales all the quantum fluctuations (in the sense of statistical processes) are irrelevant to the accuracy of our everyday observations.

 

[RUTA]

Yes, but you'll get zero by measuring the angular momentum in any direction. I guess, I now get the problem you want to discuss. Of course, you can for each single decay only measure the angular momentum in one direction, not in three linearly independent ones since angular-momentum components at different directions are not compatible to each other.

Right, the classical picture would have definite values for $\vec{L_A}$ and $\vec{L_B}$ for each of Alice and Bob's particles, respectively. $\vec{L_A}$ and $\vec{L_B}$ would have the same magnitude L and be anti-aligned along some direction in space (call that direction $\vec{d}$). When Alice and Bob make measurements of $\vec{L_A}$ and $\vec{L_B}$ along $\vec{A}$ and $\vec{B}$, respectively, they will get fractions of L correlated per conservation of angular momentum. In the quantum case, they both always measure L in every direction in such a way that Alice(Bob) can claim her(his) measurements were always along $\vec{d}$ and Bob's(Alice's) "incorrect" measurements averaged to the correct value. So, for QM there is no preferred $\vec{d}$ for this conserved vector quantity.

 

[RUTA]

As in all cases of apparent "quantum weirdness" I know, the minimal statistical interpretation resolves the weirdness. The only weirdness remaining is due to our classically trained prejudices about the behavior of objects, but these prejudices are due to our everyday experience with very much coarse-grained macroscopic observables, which are in fact averaging over many microscopic degrees of freedom, which leads to an apparent classical behavior, but in fact it's just due to the sufficiency of coarse-grained macroscopic observables to describe macroscopic systems. On these macroscopic scales all the quantum fluctuations (in the sense of statistical processes) are irrelevant to the accuracy of our everyday observations.

The weirdness is trivially resolved if you accept the QM predictions, which we know give CM via averages. That's what most physicists do, i.e., most physicists don't bother with foundations of QM. This attitude is famously called "shut up and calculate" by Mermin. As argued by Becker, physicists do require physical models to do physics (he has some nice examples in his book) and these models are what allow physicists to create new approaches to theory and experiment. Einstein thought QM was incomplete precisely because his model of physical reality would not accommodate QM predictions for entangled states. Bell's inequalities were derived precisely in response to Einstein's model of physical reality. In Sabine's new book, even Weinberg admits to looking for a theory underwriting QM because it violates his model of physical reality (that's not how he worded it of course).

What we're saying in our paper and book (and how I close my Insight) is that there is a model of physical reality (not simply "shut up and calculate" aka "instrumentalism") for which QM makes sense and is compatible with relativity. In this Insight, we see that the QM correlations follow from conservation of angular momentum for the quantum exchange of momentum as required for no preferred reference frame. That's compelling, but provides no 'causal influence' or hidden variables to account dynamically for the outcomes on a trial-by-trial basis. The constraint here only holds over space AND time, it's truly 4D, and it has no compelling dynamical counterpart. What we argue in our book (and in my blockworld Insight series) is that 4D constraints are fundamental, not dynamical laws. Most people disagree strongly with this (consider Fermat's Principle of Least Time versus Snell's Law, for example, which really explains the light ray's trajectory?). However, in case after case, we see that mysteries arise in physics because we demand dynamical explanation and all such mysteries disappear when we accept the explanation via 4D constraints. This is just one of many such examples.

So, this Insight really vindicates the shut-up-and-calculate attitude by providing a model of physical reality in which QM doesn't need to be 'fixed' or underwritten (anymore so than we already have with QFT anyway). QM is in beautiful accord with a truly 4D reality constrained in 4D fashion in such a way as to guarantee dynamical experience per CM.

I spent 24 years trying to figure out Mermin's "quantum mysteries for anybody." I finally feel as though I have the answer (a model of physical reality in which QM entanglement is in perfect accord with CM and SR). The invariant manner by which Mermin's "mysterious" QM correlations follow from conservation principles and lead to CM honestly makes me say, "how could I have been so stupid for so long?"

 

[vanhees71]

The weirdness is trivially resolved if you accept the QM predictions, which we know give CM via averages. That's what most physicists do, i.e., most physicists don't bother with foundations of QM. This attitude is famously called "shut up and calculate" by Mermin. As argued by Becker, physicists do require physical models to do physics (he has some nice examples in his book) and these models are what allow physicists to create new approaches to theory and experiment. Einstein thought QM was incomplete precisely because his model of physical reality would not accommodate QM predictions for entangled states. Bell's inequalities were derived precisely in response to Einstein's model of physical reality. In Sabine's new book, even Weinberg admits to looking for a theory underwriting QM because it violates his model of physical reality (that's not how he worded it of course).

It's of course true that you need intuitive pictures about physics to "create" (or rather "discover") new theoretical models, but Einstein is a prime example for the danger of being trapped in philosophical prejudices.

Of course, in some sense the minimal statistical interpretation indeed is indeed a kind of nicer expression for "shutup and calculate". The question is whether you can expect more from a natural science than just this: You have a model (or even theory) which allows you to predict the outcome of observations, measurements, and experiments and than compare these expectations with the observations. If these expectations agree with the data, it's fine for the model, otherwise you have to think harder about what's wrong with the model and find a new one. This is indeed a creative act, and you need intuitive pictures to get the (finally) the right idea how to describe the phenomena with existing (which is almost always the case) models/theories or you have to find a new one (this occurred only two times after Newton, i.e., with the discovery of relativity around 1905 and of quantum theory in 1925).

I know that Weinberg thinks there is something unsolved with the foundations of quantum theory from his textbook on quantum mechanics (as always among the best textbooks on the subject). Although for me Weinberg is a role model for how to do theoretical physics (with a strict "no-nonsense approach" and with a clear mathematical exposition of all the papers and textbooks by him I'm aware of), this I do not understand, since there's no contradiction whatsoever with quantum theory and its application to real-world observations. So what should be incomplete in its applications?

I've not yet read Hossenfelders new book. The title "lost in math" already appalls me, since my view on theoretical physics is the opposite (I'd rather say "lost without math" ;-)), but I think she has indeed a point in saying that maybe we have to widen our view to new (mathematical) methodology beyond the symmetry paradigm, which was indeed the right paradigm for 20th-century physics in creating quantum theory (for me there's no convincing way to formulate quantum theory without symmetry principles and Nother's works on symmetries and conservation laws), relativity, and the Standard Models of elementary particle physics and cosmology, but it may well be that we need new methods to find a unified theory of QT and GR. She is also right in saying that it is hard to conceive whether we have a chance without new empirical findings clearly contradicting one of these fundamental theories (or rather our best approximation of the maybe and hopefully existing but yet undiscovered more comprehensive theory).

Towards Becker's book, I've a mixed feeling. On the one hand I find it overdue to get Bohr, Heisenberg, et al from their pedestal. The true interpretational problem is due to the unjustified predominance of the Copenhagen flavor of interpretations, and Bohr's writings on the subject doing more harm than good, because they are usually not well formulated and too vague and too qualitative ("lost without math"! indeed) to be not subject to speculations about their meaning. That said, Heisenberg is even worse! On the other hand, I cannot agree with Becker's enthusiasm for the de Broglie-Bohm approach since there's to my knowledge no convincing formulation of relativistic QFT within this approach. Any interpretation must be an interpretation of all of the working QTs, applied to real-world phenomena, and this includes relativistic local QFT although it's still not a mathematically strictly defined theory.

What we're saying in our paper and book (and how I close my Insight) is that there is a model of physical reality (not simply "shut up and calculate" aka "instrumentalism") for which QM makes sense and is compatible with relativity. In this Insight, we see that the QM correlations follow from conservation of angular momentum for the quantum exchange of momentum as required for no preferred reference frame. That's compelling, but provides no 'causal influence' or hidden variables to account dynamically for the outcomes on a trial-by-trial basis. The constraint here only holds over space AND time, it's truly 4D, and it has no compelling dynamical counterpart. What we argue in our book (and in my blockworld Insight series) is that 4D constraints are fundamental, not dynamical laws. Most people disagree strongly with this (consider Fermat's Principle of Least Time versus Snell's Law, for example, which really explains the light ray's trajectory?). However, in case after case, we see that mysteries arise in physics because we demand dynamical explanation and all such mysteries disappear when we accept the explanation via 4D constraints. This is just one of many such examples.

Well, I've to read the Insight article again. So far I couldn't get the content of the whole approach :-(. I also do not understand, what philosophers and philosophy-attached physicists mean, when they talk about "reality". For me QT is the best description of reality we have, and the only thing that's incomplete with it is the lack of a consistent quantum description of gravity. For me there's no interpretational issue at all, and I don't think that looking for classical/deterministic non-local descriptions have a chance to lead to anything, because a non-local theory is hard to formulate within relativistic physics. One historical failure is Feynman's and Wheeler's attempt to formulate an action at a distance (non-local) theory for interacting systems of charged particles. Although this "absorber theory" seems to work to some extent on a classical level, there was (so far) nobody able to build a quantum formulation of it.

So, this Insight really vindicates the shut-up-and-calculate attitude by providing a model of physical reality in which QM doesn't need to be 'fixed' or underwritten (anymore so than we already have with QFT anyway). QM is in beautiful accord with a truly 4D reality constrained in 4D fashion in such a way as to guarantee dynamical experience per CM.

I spent 24 years trying to figure out Mermin's "quantum mysteries for anybody." I finally feel as though I have the answer (a model of physical reality in which QM entanglement is in perfect accord with CM and SR). The invariant manner by which Mermin's "mysterious" QM correlations follow from conservation principles and lead to CM honestly makes me say, "how could I have been so stupid for so long?"

QM entanglement is in perfect accord with SR and with none local classical model. So there must be a non-local aspect in what you call "classical mechanics", but as I said, I better make another attempt to understand your Insight article.