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

[RUTA]

Ok, let me try to explain Unnikrishnan's conservation principle as transparently as possible. We have two sets of data, Alice's set and Bob's set. They were collected in N pairs with Bob's(Alice's) SG magnets at $\theta$ relative to Alice's(Bob's). We want to compute the correlation of these N pairs of results which is

$\frac{(+1)_A(-1)_B + (+1)_A(+1)_B + (-1)_A(-1)_B + ...}{N}$

Now organize the numerator into two equal subsets, the first is that of all Alice's +1 results and the second is that of all Alice's -1 results

$\frac{(+1)_A(\sum \mbox{BA+})+(-1)_A(\sum \mbox{BA-})}{N}$

where $\sum \mbox{BA+}$ is the sum of all of Bob's results corresponding to Alice's +1 result and $\sum \mbox{BA-}$ is the sum of all of Bob's results corresponding to Alice's -1 result. Notice this is all independent of the formalism of QM. Now, we rewrite that equation as

$\frac{(+1)_A(\sum \mbox{BA+})}{N} + \frac{(-1)_A(\sum \mbox{BA-})}{N} = \frac{(+1)_A(\sum \mbox{BA+})}{2\frac{N}{2}} + \frac{(-1)_A(\sum \mbox{BA-})}{2\frac{N}{2}}$

which is

$\frac{1}{2}(+1)_A\overline{BA+} + \frac{1}{2}(-1)_A\overline{BA-}$

with the overline denoting average. Again, this correlation function is independent of QM formalism. All we have assumed is that Alice and Bob measure +1 or -1 with equal frequency at any setting in computing this correlation. Now we introduce our proposed conservation principle as I justified in #382 which is

$\overline{BA+} = -\cos(\theta)$

and

$\overline{BA-} = \cos(\theta)$

This gives

$\frac{1}{2}(+1)_A(-\cos(\theta)) + \frac{1}{2}(-1)_A(\cos(\theta)) = -\cos(\theta)$

which is exactly the same correlation function as the quantum correlation obtained using conditional probabilities for the spin singlet state in QM. However, again, none of the QM formalism is used in obtaining this result. In deriving the quantum correlation function in this fashion, we assumed two key things: 1) Bob and Alice measure +1 or -1 with equal frequency in any setting and 2) Alice(Bob) says Bob(Alice) conserves angular momentum on average when Bob's(Alice's) setting differs from hers(his) by $\theta$. Those two assumptions are what I mean when I say the result is "reference frame independent."

I have added this to my Insight. I also added an explicit calculation of the quantum correlation function using the conditional probabilities for the spin singlet state from QM, so you can see how the two derivations differ.

 

[vanhees71]

Can you also explain this strange notation. Already the first equation is not clear to me. What does it mean? It's an average of what? What's in the numerator? In this way it's indeed a mystery compared to quantum theory, which is not a mystery but the solution to the mystery of the observed behavior of microscopic particles as well as the then ununderstandable stability of macrocsopic matter surrounding us.

That said, let me come to your AJP preprint. I'll got through it as I'd be a referee.

Section I is confusing and doesn't make sense to me to begin with.You should explain Mermin's apparatus to make your paper self-consistent. You don't explain it but rather open several other topics (4D spacetime views and Fermat's principle) which are completely unrelated to the "conundrum of entanglement". Since Bell it's the more clear that quantum theory is not the mystery but the solution to describe the behavior of subatomic particles, in this case spin-entangled states of two particles.

As a referee, I'd suggest to cancel Sect. I and use Sect. II as the introduction, explaining clearly Mermin's apparatus. You should explain what's entangled. It's the spins of the two particles emitted from the middle box. It doesn't make sense to say "two particles are entangled" in QM. You have to say which observables are entangled. Figs. 3 and 4 are unexplained. What are they good for? To make the paper understandable to at least a physics student who has heard the QM 1 lecture, you should just explain the experiment in terms of standard QT, i.e., say that the two spin-1/2 particles are prepared in the pure $j=0$, $j_3=0$ state represented by the state vector
$$|\Psi \rangle=\frac{1}{\sqrt{2}} (|1/2,-1/2 \rangle - |-1/2,1/2 \rangle),$$
where the notation for the two-particle spin states is the usual one, i.e.,
$$|\sigma_{z1},\sigma_{z2} \rangle \equiv |\sigma_{z 1} \rangle \otimes |\sigma_{z2} \rangle.$$
To make your paper as mysterious as you can you don't even tell this your reader anywhere.

What you describe then is completely ununderstandable to me. It doesn't reflect at all what QT predicts to be measured in A's and B's measurements. It's not clearly explained. You can calculate it easily of course. You simply quote the result in Eqs. (2) and (3) without clearly saying what's measured. Obviously what's meant is that A and B choose a plane (say the $xy$ plane for simplicity since due to the total isotropy of the entangled state it doesn't matter anyway which plane they choose). Then with two unit vectors $\vec{n}(\alpha)=(\cos \alpha,\sin \alpha,0)$ what's measured are the spin components of A's and B's particles in directions $\vec{n}(\alpha)$ and $\vec{n}(\beta)$ respectively. The probabilities quoted in Eqs. (2) and (3) are then, written in standard notation
$$P(\sigma_{1\alpha},\sigma_{2\beta})=|\langle \vec{n}(\alpha) \cdot \vec{\sigma}_1,\vec{n}(\beta) \cdot \vec{\sigma}_2|\Psi \rangle|^2.$$
On the left-hand side of the equation I denoted spin components in direction $\alpha$ in the above defined sense as $\sigma_{\alpha}=\vec{n}(\alpha) \cdot \vec{\sigma})$. I'll use this abbreviation from now on.

Of course the possible outcomes for each single-particle spin component are $\pm 1/2$, and you give the correct probs. for all four possible simultaneous outcomes in Eqs. (2) to (3). But why don't you give this simple explanation rather than the very complicated description so far?

Fig. 6 and its caption is absolutely enigmatic to me. I still don't get the meaning of the words "angular momentum is conserved on average" should mean for unaligned measurements, i.e., for $\alpha-\beta \neq 0$ or $\pi$. In which sense should there be angular-momentum conservation be measured. I've brought this argument again and again already several times in this thread, and it's not answered. It doesn't even make sense in a classical context to check angular momentum conservation of a system by measuring components of angular momenta on different parts of the system in different directions! Also what's represented in this space-time diagram? Measurement outcomes of A's and B's measurements? Why do I need a space-time diagram to depict this?

I've no clue what $\langle \alpha,\beta \rangle$ should mean either. What's summed over? I can only guess it is
$$\langle 4 \sigma_{1 \alpha} \sigma_{2 \alpha} \rangle=\sum_{\sigma_{1 \alpha},\sigma_{2 \alpha} =-1/2}^{+1/2} 4 \sigma_{1 \alpha} \sigma_{2 \alpha} P(\sigma_{1 \alpha}, \sigma_{2 \alpha}).$$
Then at least I can reproduce Eq. (4).

That Alice's and Bob's "spin angular momenta cancel on average" is the next mysterious statement. Do you mean that for any single-particle spin component the average is 0? That's of course true due to the complete isotropy of the spin-singlet state. Of course, this follows also from the probabilities given by Eqs. (2) and (3). Of course, everything is completely determined by the probabilities (2) and (3). So to translate the very complicated text, what you claim is that in some way you can get these quantum probabilities by a not precisely defined "principle of angular-momentum conservation on average"? I cannot invisage how I can make sense of that, although so far I could make some conjectures about what you wanted to say. As I repeatedly said, I've no clue what the fact that in this setup the single-particle spin components have a 0 expectation value to do with angular-momentum conservation.

That's trivial for the physical situation I guessed you really want to described, given the completely isotropic preparation of the two-particle state (the $j=0$ state). Formally you get the statistics of the single-particle spins by "tracing out the other particle", and this leads to
$$\hat{\rho}=\frac{1}{2} \hat{1}=\frac{1}{2} \left (|\sigma_{\alpha}=1/2 \rangle \langle \sigma_{\alpha}=1/2| + |\sigma_{\alpha}=-1/2 \rangle \langle \sigma_{\alpha}=-1/2| \right )$$
for any $\alpha \in [0,2 \pi)$.

I'd be very interested, how your referee reports come out from AJP...

 

[RUTA]

Again, you're missing the point which is to answer Mermin's challenge to explain how his device works to the "general reader." He's able to introduce the conundrum via the Mermin device in a way accessible to the "general reader," but I wasn't able to get the explanation quite down to that level. However, I did get it down to the level of someone who completed introductory physics. So, all you need from QM to do that are the quantum probabilities for the state in question -- no Hilbert space, no density matrix, no Pauli spin matrices. The probabilities alone suffices to explain the mystery from the QM formalism. The first equation is a conventional way to write the correlation, so I'm surprised you don't recognize it. Anyway, since Unnikrishnan's conservation principle reproduces the quantum correlation (first equation plus QM probabilities), I have to translate the conundrum from probabilities to correlations. The spacetime or 4D view is necessary to justify Unnikrishnan's conservation principle as a constraint that fully resolves the conundrum, which I explain in Sec I.

 

[vanhees71]

There is no other way today to explain the behavior of matter on the fundamental level than quantum theory. That's the important result of Bell's work on local deterministic hidden-variable theories. There are also no mysteries to be resolved. You only have to accept that there are correlations in quantum physics which cannot be described by such a classical theory but are a natural consequence of quantum theory, named entanglement. These correlations can be "long-ranged", i.e., there can be correlations between properties of distinguishable parts of a quantum system which are very far away.

What's confusing in my point of view is to call this "non-locality". As Einstein already wrote in 1948 (in a paper which is much more to the point than the famous EPR paper which Einstein didn't particularly like so much) the key issue he was uneasy about was the inseparability of quantum systems through the possiblitiy of an entanglement of observables of far-distant parts of a quantum system.

Your example of the spin-entangled spin-singlet state of two-particles is paradigmatic. It's usually easier to realize with polarziation-entangled photon pairs, which nowadays are easily produced through parametric downconversion, but the principle issue is the same. In principle the polarization-entanglement can persist for arbitrary long times (as long as there's no interaction of one of the particles or photons with something else and no decoherence occurs), and thus the particles or photons can be registered by as far distant observers as one likes, and each observer can choose his observable he likes to measure (i.e., in your example which spin component he likes to measure or which polarization state he likes to filter out), but the correlations described through entanglement will be observed.

All this is fully concistent with relativistic local microcausal QFTs. For photons everything is well understood within standard quantum optics, based on QED (with the optical devices treated in hemiclassical approximation, which is of sufficient accuracy for the usual experiments). Since for QED, as for any local QFT, the linked-cluster theorem holds there are "spooky actions at a distance", but the long-ranged "stronger than classically possible" correlations are simply there because of the preparation of the two-particle/two-photon system in an entangled state. Thus although the single-particle spins (resp. single-particle photon polarizations) are maximally indetermined, there's still this strong correlation beween measurement outcomes.

Admittedly this is hard to swallow as long as you don't accept that Nature behaves as she does and doesn't care about our philosophical prejudices due to our everyday experience with macroscopic matter, which behaves pretty classical also according to QT since we don't resolve (and don't need to resolve) every microscopic detail, such that the quantum fluctuations of the corresponding macroscopic coarse-grained obserervables are practically not visible.

In your paper there's nothing explained differently from QT. All you do is to assume the probabilities of QT to be valid and then calculate expectation values due to the rules. That the average of any of the single-particle spin components in any direction is 0 is simply due to the symmetry of the sytem. That's implied by the fact that the total angular momentum is precisely 0 due to the preparation of the particle pair in this state, and this state is a maximally entangled Bell state.

Bell's brillant analysis of this state in terms of a deterministic local theory clearly shows that QT is different from any such theory, and you have to give up either locality or determinism. Since local QFTs are the most successful consistent descriptions of matter we have today in terms of the Standard Model, my personal conclusion is that we have to give up determinism, but that was known since 1926 when Born got the so far only consistent interpretation of quantum states, namely their probabilistic meaning in terms of what we now rightly call "Born's Rule".

 

[RUTA]

Again, you've missed the point entirely. Did you even read Mermin's paper? His `Mermin device' produces outcomes he calls "case (a)" and "case (b)." Case (a) outcomes obtain for like settings on his device and case (b) outcomes obtain for unlike settings. The only way he knows to explain the workings of the device in accord with case (a) outcomes, his "instruction sets," is incompatible with the case (b) outcomes, thus the conundrum. You don't need any QM to understand this conundrum, just simple probabilities. He then asks the "physicist reader" to explain how his device works to the "general reader," analogously to how he was able to explain the conundrum of the device to the "general reader." Density matrices, spin operators, and Hilbert space won't cut it. My paper is very close to meeting his challenge. In addition to simple probabilities, which are allowed, I used conservation of angular momentum, which is a bit more. Can you do better? If so, write it up and submit it!

 

[vanhees71]

I haven't missed the point. You have failed to convince me that there is a point. There's nothing non-trivial derived in your paper, and it's written in a way that one has to guess what you want to tell and there's a lot of off-topic ballast in it. Excuse me for being harsh.

 

[RUTA]

I haven't missed the point. You have failed to convince me that there is a point. There's nothing non-trivial derived in your paper, and it's written in a way that one has to guess what you want to tell and there's a lot of off-topic ballast in it. Excuse me for being harsh.

If you don't understand the conundrum, then you won't appreciate Unnikrishnan's solution and my qualification thereto. I did revise the manuscript according to my efforts to explain it to you, so these exchanges did prove useful :-)

 

[vanhees71]

Well, I've just looked up the following paper by Unnikrishnan:

DOI: 10.1209/epl/i2004-10378-y

He got the issue with the conservation law correct, i.e., precisely as I stated several times. Maybe it helps to sharpen also your manuscript if you use his explanation on pages 490 and 491 in his paper, particularly the statement on the conservation law directly under item 2) on page 491. Then it becomes really a non-trivial and interesting issue which sheds further light on Bell's inequality in showing that there's no local deterministic HV theory that obeys the angular-momentum-conservation law on average. This is weaker than to assume the conservation law to be valid for any individual system as is the case for quantum theory for the spin-singlet state.

 

[RUTA]

Well, I've just looked up the following paper by Unnikrishnan:

DOI: 10.1209/epl/i2004-10378-y

He got the issue with the conservation law correct, i.e., precisely as I stated several times. Maybe it helps to sharpen also your manuscript if you use his explanation on pages 490 and 491 in his paper, particularly the statement on the conservation law directly under item 2) on page 491. Then it becomes really a non-trivial and interesting issue which sheds further light on Bell's inequality in showing that there's no local deterministic HV theory that obeys the angular-momentum-conservation law on average. This is weaker than to assume the conservation law to be valid for any individual system as is the case for quantum theory for the spin-singlet state.

Here is his item 2:

The theory of correlations obeys the conservation of angular momentum on the average over the ensemble, and for the case of singlet state,STotal = 0, there is rotational invariance. Note that this is a weak assumption, since we do not insist on the validity of the conservation law for individual events.

He says, immediately thereafter

The second criterion is the main assumption, physically well motivated, in the proof that follows. Since the main assumption is applied only for ensemble averages and not for individual events, I do not make any explicit assumption on locality or reality.

That is exactly the point I make when I say Bob can't satisfy conservation of angular momentum on a trial-by-trial basis when he and Alice make measurements at different angles. He can only satisfy the conservation principle an average in such cases. [Of course, he can say the same about Alice.] The correlation function obtained per Unnikrishnan's conservation principle is not satisfied by "instruction sets," which is the Mermin equivalent of saying Unnikrishnan's conservation principle cannot be satisfied by any "local deterministic HV theory." Again, did you read Mermin's paper?

 

[vanhees71]

But you didn't make the point clear! Unnikrishnan does. Even under the assumption of angular-momentum conservation on average, which is less than what's the case for QT, where angular-momentum conservation holds on an event-by-event basis, he can show that there's no local deterministic HV model which leads to the violation of Bell's inequality as predicted by QT. I've not read Mermin's paper, but I don't think it's necessary, because in Unnikrishnan's paper everything is clear.

 

[RUTA]

But you didn't make the point clear! Unnikrishnan does. Even under the assumption of angular-momentum conservation on average, which is less than what's the case for QT, where angular-momentum conservation holds on an event-by-event basis, he can show that there's no local deterministic HV model which leads to the violation of Bell's inequality as predicted by QT. I've not read Mermin's paper, but I don't think it's necessary, because in Unnikrishnan's paper everything is clear.

What I had in the paper was just Unnikrishnan's summary paragraph. Obviously, I can't include all the explication he provides in his paper after that summary, but I didn't think it necessary since his summary was very clear to me. Apparently, it wasn't clear to you, so I revised the paper here replacing his summary with my "no preferred reference frame" argument for his conservation of angular momentum on average. My argument is just another way of looking at his argument or just another way of looking at Boughn's argument here. However you justify it, the key insight of Unnikrishnan is to use conservation of angular momentum on average to provide $\overline{BA+}$ and $\overline{BA-}$ in the correlation function (see post #386). That gives you the quantum correlation function without ever using quantum mechanics. This is akin to deriving the Lorentz transformations from the light postulate (in more ways than one, as I will point out).

As for articulating the fact that Unnikrishnan's result rules out "local HV theories," that's trivially clear from the fact that his conservation principle reproduces the quantum correlation function which rules out local HV theories (I have included that very statement in the paper). In the Mermin paper (had you bothered to read it), he goes to great lengths to explain how his "instruction sets" are the equivalent of any local HV theory. As with the Unnikrishnan paper, I can't include Mermin's entire paper in mine, so I must expect the reader to have read the Mermin paper. The Mermin device is a metaphor for the formalism of QM in this particular experimental set-up. So, when Mermin shows that his device cannot be explained with instruction sets, he's showing how QM rules out local HV theories. The conundrum of the Mermin device is then, "If it doesn't work via instruction sets, how the hell does it work?" Since Unnikrishnan's conservation principle gives the quantum correlation function responsible for the mysterious outcomes of the Mermin device, his conservation principle invoked as a constraint (as with the light postulate) then answers that question, i.e., resolves the conundrum of the Mermin device. However, ...

As I point out, Unnikrishnan's conservation principle only resolves the conundrum of the Mermin device if you can accept the conservation principle as a constraint on the distribution of outcomes in space and time with no `deeper mechanism' to account for the constraint proper. In other words, you have to accept the conservation principle as a constraint in and of itself without further explanation. Prima facie the conservation of angular momentum on average sounds like a perfectly reasonable constraint. But, this constraint does not provide a `deeper mechanism' at work on a trial-by-trial basis to account for the average conservation. So someone might still say, "But, what mechanism is responsible for the conservation? How do the particles `know' how to behave in each trial so as to contribute properly to the ensemble? Each particle has `no idea' what the outcomes were at both locations in preceding trials, nor does it `know' what the other device setting is in their particular trial. How the hell does this average conservation pattern in space and time get created?"

And that leads us to the other analogy with the light postulate. Even Michelson of the Michelson-Morley experiment said, "It must be admitted, these experiments are not sufficient to justify the hypothesis of an ether. But then, how can the negative result be explained?" In other words, even Michelson required some `deeper mechanism' to explain why "the speed of light c is the same in all reference frames." In general, if one cannot accept a constraint or postulate in and of itself as the fundamental explanans, that constraint or postulate is just as mysterious as the explanandum. That's the point of my paper and that is the point of our book, "Beyond the Dynamical Universe." So, my paper is just another argument for constraint-based explanation as fundamental to dynamical/causal explanation.