I Difference between preparing and measuring

[rasp]

I’m understanding that if you prepare a QM system, say 2 electrons in a singlet state and do nothing to the system then when the measurement is made the 2 electrons will still be in the singlet state. If that is true, then I don’t see anything surprising that if Bob measures his electron as up he instantaneously knows that Alices electron is down. It seems to me that the electrons were entangled in the preparation state and found to be entangled in the measurement state. This seems trivial. What am I missing?

 

[PeroK]

I’m understanding that if you prepare a QM system, say 2 electrons in a singlet state and do nothing to the system then when the measurement is made the 2 electrons will still be in the singlet state. If that is true, then I don’t see anything surprising that if Bob measures his electron as up he instantaneously knows that Alices electron is down. It seems to me that the electrons were entangled in the preparation state and found to be entangled in the measurement state. This seems trivial. What am I missing?

You're perhaps not missing anything. The difference between quantum entanglement and classical correlation is that in QM an observable does not have a well-defined value until it is measured.

If we first take a classical example: a body splits into two parts, which must have equal and opposite momenta. If you measure the momentum of one part, you know the momentum of the other. But, in the classical model, the two parts always had that specific momentum from immediately after they split.

But, in QM, any particle does not have a well-defined momentum, spin or whatever until it is measured. Now, when you add this to quantum entanglement you get the difficult issue of how nature correlates the measurements of the two particles. The two particles move in opposite directions with a total spin of zero, but neither is definitely up-spin or down-spin. But, if they are measured, the measurements always correlate. One is always up and the other down, if measured.

Interestingly, there are many threads on here where the issue of quantum entanglement is debated as, per se, something to be wondered at; whereas, you only get something unexpected when you add the tenet of quantum theory that the measurements are not determined until a measurement takes place.

 

[Nugatory]

It seems to me that the electrons were entangled in the preparation state and found to be entangled in the measurement state. This seems trivial. What am I missing?

There's no single measurement that tells us that the two electrons (strictly speaking, the single quantum system whose properties we have more or less arbitrarily categorized as "properties of one electron" or "properties of the other electron") are in the singlet state. Instead we would have to identically prepare a large number of pairs, then measure (along the same axis) the spins of both members of each pair. If our preparation procedure is in fact producing the singlet state, we will get opposite results every single time; if it is just spitting out random unentangled electrons we will get opposite results about 50% of the time. Thus, if we've only seen opposite results after measuring $N$ trials, we conclude that there is at most one chance in $2^N$ that our procedure is not producing spin-entangled pairs. (Classical analogy: If I toss a coin 100 times and it comes up heads every single time, I'm pretty confident that it is not an honest coin producing heads and tails with equal probability).

Now we can say, as you are suggesting, that the pairs are being prepared in the singlet state so there is nothing surprising about our subsequent measurements being consistent with that state. We can measure on different axes, we can try different axes for each electron, we'll confirm the $\sin^2\frac{\theta}{2}$ rule as we expect for particles prepared in the singlet state... And we get no surprises.

What is surprising (perhaps "offensive to our common sense understanding of elements of reality" would be a better term) is that the physical thing that we're actually measuring, namely the spin on various axes, cannot have been determined when the pairs were prepared. That's shown by the observed violations of Bell's inequality.

 

[rasp]

Thank-you for your responses. Unfortunately I have no experience experimenting with quantum particles, and this lack of knowledge is hindering my ability to fully appreciate the "mysteries". Let me ask if each of my assumptions are correct. Is it true that I can start with a sealed chamber and add 1 electron to the left and 1 to the right. I can apply 2 oppositely polarized magnetic fields along some axis to the electrons such that the electron on the left aligns with spin up or down and the one on the right aligns with spin down or up? Is it true that when I measure them the spins will be correlated to either <up, down> or <down, up>? But isn't that just classically consistent with my prior knowledge from the preparation? Is it true that I can be confident that if the preparation state is not disturbed prior to the measurement, then those electrons are only either<up, down> or <down, up>? Or is there some random decay that moves the electrons out of the prepared state, yet still keeps them oppositely correlated?

 

[PeroK]

Thank-you for your responses. Unfortunately I have no experience experimenting with quantum particles, and this lack of knowledge is hindering my ability to fully appreciate the "mysteries". Let me ask if each of my assumptions are correct. Is it true that I can start with a sealed chamber and add 1 electron to the left and 1 to the right. I can apply 2 oppositely polarized magnetic fields along some axis to the electrons such that the electron on the left aligns with spin up or down and the one on the right aligns with spin down or up? Is it true that when I measure them the spins will be correlated to either <up, down> or <down, up>? But isn't that just classically consistent with my prior knowledge from the preparation? Is it true that I can be confident that if the preparation state is not disturbed prior to the measurement, then those electrons are only either<up, down> or <down, up>? Or is there some random decay that moves the electrons out of the prepared state, yet still keeps them oppositely correlated?

That doesn't seem to be very well aligned with the concept of entanglement. Try this:

https://www.pbs.org/video/pbs-space-time-entanglement/

 

[DarMM]

Interestingly, there are many threads on here where the issue of quantum entanglement is debated as, per se, something to be wondered at; whereas, you only get something unexpected when you add the tenet of quantum theory that the measurements are not determined until a measurement takes place.

Well that's only one take on it, an interpretation. In several interpretations the measurements are determined in advance.

Entanglement is confusing/surprising as you cannot simulate it with a local non-retrocausal single world physical theory in which spacetime is not topologically highly nontrivial (this is Bell's theorem), a fairly surprising result I think for a two electron system.

 

[DarMM]

@rasp , I'll illustrate with something that is close to quantum entanglement. Technically its more strongly correlated than you'd get in QM, but it's easier to understand. At the end I'll mention the small modifications needed to make it exactly like QM.

I have one photon and I can measure its polarization in one direction, X, and in a direction at 45 degrees to it, Z. I also have a second photon with which I can do the same. So that's four variables: $X_1, Z_1, X_2, Z_2$. If you don't know much about polarization all that matters is that when you measure these values, they can only be $0$ or $1$.

Now when I do observations I find that they obey the following conditions:
$$
X_1 = X_2\\
X_1 = Z_2\\
Z_1 = X_2\\
Z_1 \neq Z_2
$$
Try to think what values each of those variables could have had before measurement to obey these conditions.

 

[Nugatory]

But isn't that just classically consistent with my prior knowledge from the preparation? Is it true that I can be confident that if the preparation state is not disturbed prior to the measurement, then those electrons are only either<up, down> or <down, up>?

Well, not all forms of "add electron from the left and 1 from the right" will lead to an entangled pair in the singlet state. But if you do have the singet state, then the answers are "yes" and "yes" as long as you are measuring both electrons on the same axis. The results are completely consistent with one electron being spin up on that axis and the other one being spin down on that axis.

The problem only starts to appear when you measure them on different axes, say 20 degrees apart. The correlations that are observed are inconsistent with any determination of the spins at pair creation time. This is bell's theorem, and you will want to study it some before you continue the discussion. There are some good references at http://www.drchinese.com/Bells_Theorem.htm (maintained by our own @DrChinese) and despite the misleading subtitle https://static.scientificamerican.com/sciam/assets/media/pdf/197911_0158.pdf is good.

Or is there some random decay that moves the electrons out of the prepared state, yet still keeps them oppositely correlated?

It's more that the prepared state is the singlet state, and the singlet state says only that if you measure the spins in aparticular ways you will get particular correlations. It doesn't give the unmeasured spins any value, so there's nothing to move out of.

 

[rasp]

Thank you so much for your replies. Unfortunately at this stage of my education I am limited in understanding some of your comments. However, I would still appreciate if you comment once again on a conclusion I reached after watching the PBS video recommended by perok.
My conclusion is that the prepared entangled singlet state does not persist in time along the prepared axis but when measured can be found in any number of directions, probablistically. Therefore, 1 quantum mystery, so to speak, is that if these electrons were physical independent and able to evolve randomly, then why does a Measurement found in the direction of the first electron result in the knowledge of a corresponding change in the other electron. Please let me know if I’m starting to understand the “strangeness” of QM.

 

[kuruman]

My conclusion is that the prepared entangled singlet state does not persist in time along the prepared axis but when measured can be found in any number of directions, probablistically.

I think you got it backwards. When the two electrons are in the singlet state, there is a prepared state, but there is no prepared axis. The singlet state doesn't "know" which way you, in the lab, have oriented the z-axis of your spin-measuring machine. The results of the measurement will be "+z" or "-z", the z-direction defined by your machine, not by a preexisting axis. It is incorrect to say that "... when measured (it) can be found in any number of directions, probablistically." When measured, it can be found either along the +z direction or in the -z direction as defined by your machine.

 

[Nugatory]

My conclusion is that the prepared entangled singlet state does not persist in time along the prepared axis

There never was a preferred axis if you have the singlet state.

Therefore, 1 quantum mystery, so to speak, is that if these electrons were physical independent and able to evolve randomly...

They aren't - that why I made that parenthetical "strictly speaking" comment in post #3 above. As far as the mathematical formalism is concerned you have a single quantum system that is connected to two spin-measuring devices and the only thing you can take to the bank is that if they're aligned on the same axis one will read "electron up here" and the other will read "electron down here". It feels like an easy and obvious step from there to "we had two electrons all along, and one was up and the other was down", but that step is justified by neither the math nor the actual measurements - it requires additional assumptions about what is going on when there are no measurements.

then why does a Measurement found in the direction of the first electron result in the knowledge of a corresponding change in the other electron?

Why must there have been a change in the "other electron" (that's in scare-quotes because as the above suggests, it's a dubious concept)? For all we know, the other electron was measured first and it was spin-up, so of course and unsurprisingly our measurement is spin-down. The only thing we know for sure is that we got one result and if someone at the other detector performs a measurement and we later get together with them and compare notes, we'll find that our result is the opposite of theirs.

 

[rasp]

I'm nearly embarrassed to ask 1 more question, but hopefully the answer will help clarify the underlying concepts for me. If I prepare 2 electrons in the singlet state, what simple measurement can I do on them which will yield a different value than expected if the 2 electrons obeyed classical logic?

 

[DrChinese]

IIf I prepare 2 electrons in the singlet state, what simple measurement can I do on them which will yield a different value than expected if the 2 electrons obeyed classical logic?

Nugatory actually explained this in post #8, once you re-read it.

Bell's Theorem shows that the quantum prediction does not allow for the classical assumption of separability of the 2 electrons. This is not obvious when the electrons are in the singlet state and close to each other, but becomes clearer as you separate them. It also requires that you look at spin angle differences other than 0 and 90 degrees. Instead: They operate as a single system consisting of 2 particles, and their statistics cannot be properly represented as Product state statistics (2 independent particles). It is "as if" a measurement on one affects the other (and vice versa). There are a variety of ways to interpret the non-classical results.

 

[Swamp Thing]

You might find this interesting https://faraday.physics.utoronto.ca/PVB/Harrison/BellsTheorem/Flash/Mermin/Mermin.swf

 

[stevendaryl]

I’m understanding that if you prepare a QM system, say 2 electrons in a singlet state and do nothing to the system then when the measurement is made the 2 electrons will still be in the singlet state. If that is true, then I don’t see anything surprising that if Bob measures his electron as up he instantaneously knows that Alices electron is down. It seems to me that the electrons were entangled in the preparation state and found to be entangled in the measurement state. This seems trivial. What am I missing?

Nothing, so far. What you've described is perfectly consistent with a classical hidden variables model:

1. Assume that each electron has an associated unit spin vector $\overrightarrow{S}$.
2. Assume that in the singlet state, the spin of one electron is the negative of the other.
3. Assume that if an electron has spin vector $\overrightarrow{S}$ and you measure the spin along a different vector, $\overrightarrow{S'}$, then you will get spin-up if the angle between the two vectors is less than 90^o, and you will get spin-down if it is greater than 90^o

This model completely explains the perfect anti-correlation between spin measurements along the same axis: It predicts that if you measure one electron of a singlet along axis $\overrightarrow{S'}$, then you will get the opposite result from measuring the other electron along that axis.

However, it does not correctly reproduce the statistics for measuring the two electrons along two different axes. Bell proved that there is no model that can reproduce the predictions of Quantum Mechanics in this case for all choices of measurement axes.

 

[stevendaryl]

I'm nearly embarrassed to ask 1 more question, but hopefully the answer will help clarify the underlying concepts for me. If I prepare 2 electrons in the singlet state, what simple measurement can I do on them which will yield a different value than expected if the 2 electrons obeyed classical logic?

Here's a concrete example: Let's pick three different axes: A, B, C, with A being the y-axis, B being 120 degrees clockwise away from A in the x-y plane, and C being 120 degrees clockwise away from B in the x-y plane.

Let's assume that we have the following perfect correlation condition: If both electrons are measured along axis A, the results will always be the opposite.

Classically, the only way to guarantee that both measurements always return opposite results is if the results are pre-determined. That is, associated with each electron is a triple of values $(a, b, c)$, where $a, b, c$ are either +1 (for spin-up) or -1 (for spin-down), with the interpretation that if you measure along axis A, you'll get result $a$, and if you measure along B, you'll get $b$, and if you measure along C, you'll get $c$. The other electron has to have the opposite numbers, $(-a, -b, -c)$.

So there are 8 possible "states" of an electron that are relevant for this situation:

1. $a=1, b=1, c=1$
2. $a=1, b=1, c=-1$
3. $a=1, b=-1, c=1$
4. $a=1, b=-1, c=-1$
5. $a=-1, b=1, c=1$
6. $a=-1, b=1, c=-1$
7. $a=-1, b=-1, c=1$
8. $a=-1, b=-1, c=-1$

So now we can prove an inequality related to Bell's inequality. Let $P(a,b,c)$ be the probability that the first electron has state $(a,b,c)$ and the second electron has state $(-a,-b,-c)$.

Define the following events:

1. The first electron is spin-up along axis A, and the second electron is spin-down along axis B. In terms of our states above, there are 2 ways this can happen: (1) The first electron is in state $(+1, +1, +1)$ and the second is in the state $(-1, -1, -1)$. (2) The first electron is in state $(+1, +1, -1)$ and the second is in state $(-1, -1, +1)$. The probability of this is $P_1 = P(+1,+1,+1) + P(+1, +1, -1)$
2. The first electron is spin-down along axis B and the second electron is spin-up along axis C. Using the same sort of reasoning we can see that the probability of this is $P_2 = P(+1, -1, -1) + P(-1, -1, -1)$.
3. The first electron is spin-up along axis A, and the second electron is spin-up along axis C. The probability of this is $P_3 = P(+1, +1, -1) + P(+1, -1, -1)$

So $P_1 + P_2 = P(+1,+1,+1) + P(+1, +1, -1) + P(+1, -1, -1) + P(-1, -1, -1)$
(rearranging) $= P(+1, +1, -1) + P(+1, -1, -1) + P(+1, +1, +1) + P(-1, -1, -1)$
$= P_3 + P(+1, +1, +1) + P(-1, -1, -1)$

As long as $P(+1, +1, +1) + P(-1, -1, -1) \geq 0$ (which must be the case, since they are probabilities, which can't be negative), we conclude:

$P_1 + P_2 \geq P_3$

But as a matter of fact, the prediction of quantum mechanics for this experiment is:

$P_1 = 1/8$
$P_2 = 1/8$
$P_3 = 3/8$

So the quantum predictions are not compatible with this type of model.