# Is this a "fair" description of entanglement for a student?

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• jon4444

#### jon4444

I'd like to have a simple way of describing quantum entanglement for high school students and would appreciate feedback on whether the following is accurate (enough):
"One type of entanglement involves spin, where two entangled particles will always have opposite spin (coming from conservation of angular momentum). The wave function tells you the probability of a single particle's spin at any point in time. You repeatedly measure one particle's spin and it obeys the wave function, but the second particle's spin, so long as it's measured after the first particle is measured, will always have opposite spin from the first (i.e., it won't obey the probabilities given by the wave function).

This description still lends itself to the question of whether the particles simply had some specific pair of opposite spins before measurement, which a sufficient amount of entanglement actually rules out (by violating Bell inequalities, for example)

If you've already talked about the spin of a single particle, how its spin state can somehow be part spin-up $|\uparrow\rangle$ and part spin-down $|\downarrow\rangle$ at the same time, such as the state $|s\rangle = (1/\sqrt{2})(|\uparrow\rangle + |\downarrow\rangle)$, then entanglement is what you get when you apply this quantum principle (the principle of superposition) to a pair or a group of particles.

If a pair of particles 1 and 2 is in a spin state like $|s_{12}\rangle = (1/\sqrt{2}) (|\uparrow_{1} , \downarrow_{2}\rangle - |\downarrow_{1}, \uparrow_{2}\rangle)$ this state is entangled because you cannot separate it into just a state of particle 1 times the state of particle 2. Entanglement is fundamentally the inseparability of joint quantum states into states of individual particles. It certainly has a lot of good applications, though:)

Greg Bernhardt
"...still lends itself to the question of whether the particles simply had some specific pair of opposite spins before measurement.."
I would have thought this possibility is disproved by showing the differences in probabilities--i.e., measure one of the particles at a specific point in time after separation and you get one probability distribution (which is described by the wave function). Conduct the same experiment on the same particle at the same time, but this time first measure the other particle--now you get a different probability distribution describing the measurements on the "first" particle even though the only change was a measurement made on the distant second particle.

"...still lends itself to the question of whether the particles simply had some specific pair of opposite spins before measurement.."
I would have thought this possibility is disproved by showing the differences in probabilities--i.e., measure one of the particles at a specific point in time after separation and you get one probability distribution (which is described by the wave function). Conduct the same experiment on the same particle at the same time, but this time first measure the other particle--now you get a different probability distribution describing the measurements on the "first" particle even though the only change was a measurement made on the distant second particle.

Unfortunately, the probability distributions are symmetric between particles. If you condition on a measurement outcome of particle 1, the conditional probability distribution of particle 2 will show outcomes opposite to the outcome of particle 1, but this is the same the other way around. This is independent of when the particles are measured.

One way (and there are many) to prove that the particles are entangled is to show that particles 1 and 2 are highly correlated not just in spin-up/down, but also in spin-left/right.
A particle in the spin-left or spin-right state is an equal superposition of spin-up and spin-down, and vise versa. If we assume the effect of measurement cannot travel faster than light, then the uncertainty principle forbids the pair of particles from having strong correlations in both these directions. Demonstrating that they actually do would demonstrate the EPR paradox, and therefore prove the entanglement.

jon4444
When discussing entanglement, I believe it is necessary to included in the description or discussion the concept of non-locality, which is the heart of Bell's Theorem and the EPR thought experiment. Once the issue of non-locality is understood, as conceived by Bell's Inequalities, quantum entanglement can be understood. I guess one can say that there is entanglement when there is a violation of Bell's Inequalities.

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I guess one can say that there is entanglement when there is a violation of Bell's Inequalities.
Yes, but the converse is not true.

The wave function tells you the probability of a single particle's spin at any point in time.

Not for the case you describe. For the case you describe, the wave function does not describe two single particles; it describes one two-particle system. In other words, it tells you the probabilities for how correlated spin measurements on the two particles will be, as a function of the directions chosen for each spin measurement. In the particular case of spin measurements both chosen in the same direction, the wave function says the results will always be opposite; but there are an infinite number of ways of choosing the relationship between the directions of the two spin measurements, and the wave function gives you predicted correlations for all of them.

the second particle's spin, so long as it's measured after the first particle is measured, will always have opposite spin from the first (i.e., it won't obey the probabilities given by the wave function)

This is wrong. See above.

jon4444
I'd like to have a simple way of describing quantum entanglement for high school students and would appreciate feedback on whether the following is accurate (enough):
"One type of entanglement involves spin, where two entangled particles will always have opposite spin (coming from conservation of angular momentum). The wave function tells you the probability of a single particle's spin at any point in time. You repeatedly measure one particle's spin and it obeys the wave function, but the second particle's spin, so long as it's measured after the first particle is measured, will always have opposite spin from the first (i.e., it won't obey the probabilities given by the wave function).

Two critical points:

When two particles are entangled they share a wavefunction, which describes a two particle system.

Measuring the particles results in a correlation between the measurements in accordance with the shared wave-function.

Try a metaphor. Bob and Alice will go to two different locations and be asked one question from a list of questions. They may not be asked the same question. They must answer the questions according to a set of instructions we give them.

In the case of entanglement, we give them each a copy of instructions that specify a joint behavior. For example, the questions might be:

1) Do you like dogs?
2) Do you like cats?
3) Do you like beer?

A set of instructions specifying joint behavior has commands like:

... etc.

Obviously Bob and Alice cannot follow such instructions unless they communicate with each other to know what the other person is being asked. This can be contrasted with "unentangled" instructions, where Alice and Bob are given instructions about what each must do as an individual, independently of what the other person does.

One could embellish this metaphor by introducing probability. We prepare a stack of papers, each with a different set of joint instructions. We pick a paper at random and give each of Alice and Bob a copy.

Results like Bell's inequality show that the statistics (of answers to questions) produced by randomly selecting joint instructions, can't always be duplicated by giving each of Alice and Bob randomly selected "unentangled" instructions, no matter how we randomly select those unentangled instructions. (You could give the class a metaphorical version of the @DrChinese challenge by having them propose unentangled instructions that produce the same statistics as various entangled instructions.)

The unexplained mystery of entanglement is that collections of pairs of physical objects behave (statistically) as if each pair is following a set of joint instructions without any means of communication being evident.

DrChinese, jon4444, richrf and 1 other person
Is this metaphor appropriate? If so, to me, it is engagingly spooky.

Proposed as an illustration of what entanglement implies (it does not do a good job of illustrating superposition I don't think) -

I put two spinning coins in separate opaque boxes and I do not keep track of any information regarding heads / tails on the coins. I decide when to press a button on box A that stops the coin from spinning. Once I look at coin A to assess if it stopped at either heads or tails, I can press the button on box B any time I want to, and no matter when I press it, I am assured beforehand what the result is going to be, even though the coin in box B is still spinning until I press the button. Its still spinning, but no matter when I choose to stop it, the result is preordained.

Is this metaphor appropriate? If so, to me, it is engagingly spooky.

Proposed as an illustration of what entanglement implies (it does not do a good job of illustrating superposition I don't think) -

I put two spinning coins in separate opaque boxes and I do not keep track of any information regarding heads / tails on the coins. I decide when to press a button on box A that stops the coin from spinning. Once I look at coin A to assess if it stopped at either heads or tails, I can press the button on box B any time I want to, and no matter when I press it, I am assured beforehand what the result is going to be, even though the coin in box B is still spinning until I press the button. Its still spinning, but no matter when I choose to stop it, the result is preordained.

The missing element of your metaphor: What are you measuring? There are a large (or infinite) number of things to measure on A. Once you select WHAT you are measuring, the other (B) will seemingly take on an appropriate value to match. I say seemingly because: there is no sense in which you can say B becomes like A more than you can say A becomes like B. Other than by assumption or convention, of course.

And we know further, from Bell's Theorem, that there is no preordained instruction set for A and B that can explain this.

bhobba
Thanks very much for the feedback.

There are a large (or infinite) number of things to measure on A.

I meant to invoke an image of a system that has one property of interest, and two possible values for that property, heads and tails being the two values, and the property I didn't name at all, but might be "face". Did I do a poor job of articulating, or am I flatly missing something in the description I think I am accomplishing with the metaphor?

there is no sense in which you can say B becomes like A more than you can say A becomes like B.

Is this captured by saying that one can measure either A first / B second or B first / A second and show equivalent entanglement between A and B or is there something more?

[..]
Is this captured by saying that one can measure either A first / B second or B first / A second and show equivalent entanglement between A and B or is there something more?
Yes, the entanglement is captured by the fact that the entangled pair always have the correlation or anti-correlation in the entangled property whatever other circumstances prevail. There's nothing in the equations beyond that.

1. I meant to invoke an image of a system that has one property of interest, and two possible values for that property, heads and tails being the two values, and the property I didn't name at all, but might be "face". Did I do a poor job of articulating, or am I flatly missing something in the description I think I am accomplishing with the metaphor?

2. Is this captured by saying that one can measure either A first / B second or B first / A second and show equivalent entanglement between A and B or is there something more?

1. I get that you are picturing a single (identical) measurement on the system. However, that picture is so limited as to be deceptive. Specifically: if A is measure on basis X (1 or many possible measurement bases), then B acts as if it has been measured on the exact same basis X. That's an incredible coincidence, unless there is some kind of nonlocal action occurring.

2. Yes and no. The sequence of measurements is (observationally) irrelevant. Therefore, there is no way to ascribe causal ordering (if there is any action occurring).

bhobba and Grinkle
When entanglement is discussed, what is the physical origin of the particles? And is entanglement only ever regarding two particles, or can any number of particles be considered entangled? Also, can different types of particles be entangled, for example an electron and some other particle that isn't an electron?

When entanglement is discussed, what is the physical origin of the particles? And is entanglement only ever regarding two particles, or can any number of particles be considered entangled? Also, can different types of particles be entangled, for example an electron and some other particle that isn't an electron?

The physical origin of entangled particles varies widely, there is nothing specific I can think of to consider.

There is no specific limit to the number N of particles that are entangled, and N can be a very large number. However, the entangled effects tend to be progressively less dramatic anytime N is greater than 2 or 3.

And yes, different types of particles altogether can be entangled. There is no specific limit to "what" can be entangled, and in fact there are hundreds of permutations that I have read about. Many are objects of the same type, but they can be basic particles, atoms, molecules, etc. An example might be an electron and an molecule with a + charge (ion).

The physical origin of entangled particles varies widely, there is nothing specific I can think of to consider.

There is no specific limit to the number N of particles that are entangled, and N can be a very large number. However, the entangled effects tend to be progressively less dramatic anytime N is greater than 2 or 3.

And yes, different types of particles altogether can be entangled. There is no specific limit to "what" can be entangled, and in fact there are hundreds of permutations that I have read about. Many are objects of the same type, but they can be basic particles, atoms, molecules, etc. An example might be an electron and an molecule with a + charge (ion).
I guess that blows my idea out of the water then. I was thinking that maybe the entangled particles only appear to be two distinct particles in three dimensions, but in a higher dimension are actually the same particle. For example if particles were actually 4D toruses (in some sense) then to us 3Ders the single 4d particle would appear to be two distinct 3d particles and I'd think properties of these "two" particles would of course be related because they're actually properties of a single 4d particle. Guess not.

if particles were actually 4D toruses (in some sense) then to us 3Ders the single 4d particle would appear to be two distinct 3d particles and I'd think properties of these "two" particles would of course be related because they're actually properties of a single 4d particle. Guess not.

Please review the PF rules for personal speculation. This bears no resemblance to any physical model that anyone is studying in this area.

bhobba
Please review the PF rules for personal speculation. This bears no resemblance to any physical model that anyone is studying in this area.

Indeed.

Bell etc is one of those areas people get really confused about reading popularisations.

Here is all that going on - it's all in Bells original paper:
https://hal.archives-ouvertes.fr/jpa-00220688/document

Its just a correlation with different statistical properties than classical correlations. This is hardly surprising since QM is a generalization of ordinary probability theory:
https://arxiv.org/abs/1402.6562

It's hardly surprising it will have different statistical properties for correlations - Bell proved indeed it did and it was verified experimentally. Want it to be the same as ordinary probability theory for such correlations? If you insist you can do it - but then non-local interactions are required.

Personally I have a bit of a 'maverick' view of it all - I don't believe thinking in terms of correlation and locality is the right way to go because of what is known as the cluster decomposition property:

As you can see for this simple intuitive property of Quantum Field Theory its easiest to exclude correlations from considerations of locality. Of course you can still keep it and run into all the issues people talk about with Bell - its up to you - but for me simple is less confusing.

Thanks
Bill

vanhees71
Indeed.

Bell etc is one of those areas people get really confused about reading popularisations.l
Here's an issue I'm left unsure about after reading one of those popularizations ("The Order of Time"). Why didn't Einstein see the implications of entanglement with polarization that Bell saw? Was it that those aspects of polarization hadn't been worked out yet, or Einstein somehow just missed the idea of the experiments that eventually confirmed Bell's take?
(Because the experimental setups which disproved possibility of hidden variables seem fairly straightforward, once the nature of the probabilities is laid out---what did Einstein miss?)

Here's an issue I'm left unsure about after reading one of those popularizations ("The Order of Time"). Why didn't Einstein see the implications of entanglement with polarization that Bell saw? Was it that those aspects of polarization hadn't been worked out yet, or Einstein somehow just missed the idea of the experiments that eventually confirmed Bell's take?
(Because the experimental setups which disproved possibility of hidden variables seem fairly straightforward, once the nature of the probabilities is laid out---what did Einstein miss?)

The experiments require a spark of inspiration. Not unlike SR, once you see it, you think why didn't someone think of that before? But, no one did.

bhobba and jon4444
Historically, it was Bohm's causal formulation, which Van Neumann formerly "proved" was impossible, that was the inspiration for Bell. Bohmian quantum theory explicitly embraced non-local, instantaneous action at a distance, via the Quantum Potential. It was this paper and Bohm's idea to use spin that inspired Bell. Everyone else ignored Bohm's papers for various poltucal, academic reasons, even Einstein.

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jon4444
Historically, it was Bohm's causal formulation, which Van Neumann formerly "proved" was impossible, that was the inspiration for Bell. Bohmian quantum theory explicitly embraced non-local, instantaneous action at a distance, via the Quantum Potential. It was this paper and Bohm's idea to use spin that inspired Bell. Everyone else ignored Bohm's papers for various poltucal, academic reasons, even Einstein.

That is as may be, but the simple answer is that hidden variables would produce traditional probabilities, while QM predicts probabilities based on complex amplitudes. Bell's inequality neatly exploits that difference.

I honestly don't see where Bohmian mechanics comes into the argument.

Here's an issue I'm left unsure about after reading one of those popularizations ("The Order of Time"). Why didn't Einstein see the implications of entanglement with polarization that Bell saw? Was it that those aspects of polarization hadn't been worked out yet, or Einstein somehow just missed the idea of the experiments that eventually confirmed Bell's take?
(Because the experimental setups which disproved possibility of hidden variables seem fairly straightforward, once the nature of the probabilities is laid out---what did Einstein miss?)

That is as may be, but the simple answer is that hidden variables would produce traditional probabilities, while QM predicts probabilities based on complex amplitudes. Bell's inequality neatly exploits that difference.

I honestly don't see where Bohmian mechanics comes into the argument.

There was a question why Einstein didn't notice what Bell noticed. I was putting it in historical context. Bell noticed it because he did not carry the biases of other physicists at the time. He actually read Bohm's papers and that is where he received his inspiration for how to experimentally test for non-locality.

Was it that those aspects of polarization hadn't been worked out yet?

Maybe it was because most popularizations are wrong, as they are on many things in QM.

It is a misnomer to believe Einstein did not understand QM or believe in not correct - he believed it a correct theory - but just like Classical Mechanics, was incomplete - Classical Mechanics is incomplete because it leads to inconsistencies such as the Rayleigh-Jeans law where Blackbody radiation blew up to infinity and other stuff like the Lorentz-Dirac equation that has acausal runaway solutions. He thought he had found an inconsistency in QM meaning there must be a deeper theory.

Now a very great polymath called Von-Neumann had a theorem that said no hidden variables could exist. It was based on a possibly false assumption - called non-contextuality. This assumption was spotted by Grette Hermann, but in a very sad part of science she was ignored. Everyone, possibly even including Einstein (Atty thinks below he may have known about it), believed the theorem. But later Bell looked at it again, because a new theorem by a guy called Gleason explicitly used the assumption (ie non-contextuality) to basically prove what's called the Born Rule. It showed it was the hidden assumption of Von-Neumann and Bell spotted it, so he embarked on figuring out how to test it. What he showed is, basically, you can have hidden variables, if you want it (ie you can have contexuality), but you must have non-local influences of some sort.

As I explained my view is for correlations locality is not an issue - correlations are not part of it.

The following explains some of the history - and yes Bohmian Mechanics did play a role as well:
https://plato.stanford.edu/entries/bell-theorem/

Thanks
Bill

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For a more detailed description of the history, which accurately describes Bohm's central role in the development of current foundational quantum theory, I invite forum members to read Adam Becker's "What is Real". Extremely well researched and documented. His talk on the subject is available on YouTube.

That is as may be, but the simple answer is that hidden variables would produce traditional probabilities, while QM predicts probabilities based on complex amplitudes. Bell's inequality neatly exploits that difference.

I honestly don't see where Bohmian mechanics comes into the argument.

Bohmian mechanics was historically central to the development of the Bell inequalities.

Historically, it was Bohm's causal formulation, which Van Neumann formerly "proved" was impossible, that was the inspiration for Bell. Bohmian quantum theory explicitly embraced non-local, instantaneous action at a distance, via the Quantum Potential. It was this paper and Bohm's idea to use spin that inspired Bell. Everyone else ignored Bohm's papers for various poltucal, academic reasons, even Einstein.

Einstein did not ignore Bohm's papers. He just thought the solution was not beautiful (and that a more beautiful hidden variables theory could be obtained). In fact, Einstein was central to the development of Bohmian mechanics. Bohm had made the erroneous claim that quantum mechanics is inconsistent with hidden variables in his textbook. Einstein pointed out to Bohm that the claim was based on wrong reasoning, leading Bohm to eventually develop Bohmian mechanics.

bhobba
Now a very great polymath called Von-Neumann had a theorem that said no hidden variables could exist. It was based on a possibly false assumption - called non-contextuality. This assumption was spotted by Grette Hermann, but in a very sad part of science she was ignored. Everyone, including Einstein, believed the theorem.

I'm skeptical that Einstein believed wrong interpretation of the theorem. When Bohm claimed in his textbook that hidden variables were impossible, Einstein pointed out the flaw in Bohm's reasoning.

(i.e., it won't obey the probabilities given by the wave function).

This statement is wrong. The probabilities will be those given by the wave function. After the measurement of one spin, the wave function is altered so that it correctly predicts the probabilities observed when the second spin is subsequently measured.

I'm skeptical that Einstein believed wrong interpretation of the theorem. When Bohm claimed in his textbook that hidden variables were impossible, Einstein was one of those who pointed out the flaw in Bohm's reasoning.

I don't know if his reasoning was Von-Neumann's. Yes - its possible he knew it was bollocks - but I haven't read he knew the flaw. Greta was ignored - but could they ignore Einstein?

Thanks
Bill

Einstein did not ignore Bohm's papers. He just thought the solution was not beautiful (and that a more beautiful hidden variables theory could be obtained). In fact, Einstein was central to the development of Bohmian mechanics. Bohm had made the erroneous claim that quantum mechanics is inconsistent with hidden variables in his textbook. Einstein pointed out to Bohm that the claim was based on wrong reasoning, leading Bohm to eventually develop Bohmian mechanics.

Yes - along with many things about Einstein explained in Subtle is the Lord. That's why I didn't think Einstein knew the flaw because its not mentioned there.

Thanks
Bill

I don't know if his reasoning was Von-Neumann's. Yes - its possible he knew it was bollocks - but I haven't read he knew the flaw. Greta was ignored - but could they ignore Einstein?

I'm not sure what the exact history is about Von Neumann's wrong interpretation, Grete Hermann and Einstein, but here it says that Einstein pointed out the flaw in Bohm's claim in his textbook that hidden variables are impossible: http://www.bohmianmechanics.org/background/history-of-bohmian-mechanics.html.

Also, the standard 1961 textbook of Messiah does say that hidden variables cannot be ruled out and associates the hidden variable programme with Einstein. Messiah goes on to use Copenhagen (for the practical purpose of doing quantum mechanics) without ruling out hidden variables.

Also, the standard 1961 textbook of Messiah does say that hidden variables cannot be ruled out and associates the hidden variable programme with Einstein. Messiah goes on to use Copenhagen (for the practical purpose of doing quantum mechanics) without ruling out hidden variables.

Local hidden variables were not so clearly ruled out in 1961, so the Einstein program could have been considered viable at that time.

PeroK