# How do particles become entangled?

1. Nov 25, 2004

### TheDonk

How do particles become entangled? I've heard that it's when two particles bump into each other. How is this "bump" defined? What does it mean for 2 particles to bump? Is it based on distance apart, or something else?

2. Nov 26, 2004

### marlon

3. Nov 26, 2004

### TheDonk

I don't think it does...

"Well let's say that Alice and Bob both deliver a particle (A and B) and that we get an entangled state."

4. Nov 26, 2004

### jvangael

I know for example in one of the first experiments performed to test the Bell inequalities, Alain Aspect used a phenomenon which is called Atomic Radiative Cascades to create entangled photons. The basic idea was to excite a certain atom (Calcium?) to an excited state. Most atoms fall back in a lower state sending out one photon, but once in a while the atom makes a "pitstop" at an intermediate level and thus sends out 2 photons. Using some angular momentum conservation you can show that these two photons are entangled. I don't know the exact details but this is what I recall.

Hope it helps,

Jurgen

5. Nov 26, 2004

### caribou

I'm not sure of what the "bump" is either, only that it's an interaction and entanglement is extremely common in nature from all the interactions going on.

6. Nov 26, 2004

### NanoTech

Marlon's analogy is good,but I am a little confused about how the density matrix would determine that there is no measurement that particle A can perform in order to distinguish the two ensembles, maybe I need to look at it from a relativistic stand-point of each particle. Dave

7. Nov 27, 2004

### blip

My brain is still trying to wrap itself around exactly what happens during entranglement but let me ask two questions and see if someone (like Marlon or otherwise) can help me.

Question 1

Let´s assume Bob and Alice are holding each of the respective entangled photons and Bob goes zooming off into space.
At this point neither of them know what orientation the photons are in right? (Or does that depend on the source of their creation, like a calcite crystal?). Anyhow, when Bob measures his I understand that there is no way for Alice to know the measurement with more than 50% accuracy. But can she know whether he's made any measurement at all? Making the 1 bit NO MEASUREMENT, and the 0 state MEASUREMENT.

Questions 2

After the initial measurement between a pair of entangled photons the wave function has collasped and any further measurements won't influence the other, right?

8. Nov 27, 2004

### jvangael

Hi Blip,

So when Bob performs a measurement Alice does not gain any information on her state nor can she tell whether Bob made his measurement. TBut, Bob can conclude in which state Alice's qubit will be in when she measures.

After Alice and Bob's measurements are performed on their entangled pair, the wave function indeed collapses. Because Bob knows which state his qubit is in for sure and Alice knows which state her qubit is in for sure, the state of their qubits is a product state or 2 unrelated physical systems or qubits.

Hope it helps.

Jurgen

9. Nov 27, 2004

### blip

Thanks jvangael for the response, it does help.

10. Nov 27, 2004

### alexepascual

TheDonk must be frustrated by now. None of the replies really answers his question. I was going to post exactly the same question when I found your post. I'll re-formulate the question to see if this helps and I hope it agrees with the original question's intent.
I would assume we know what entanglement means in a context such as EPR, quantum teleportation, etc. So, this is not the problem. Most descriptions of entanglement describe features of two particles that show that they are indeed entangled, and the more quatitative analyses may describe the degree of entanglement by using the density matrix, but this is not the question we are seeking an answer for either.
Let's say we have a hydrogen molecule. We know from Pauli's exclusion principle that the electrons will have opposite spin. If we separate the atoms, their spins will be entangled. So far so good. But the question is: How did the electrons become entangled when the molecule was created? By what mechanism did the states where both spins are "down" or both are "up" dissapear?.
A similar example could be given in terms of a collision. We heard that every time that two particles interact they become entangled. So if two particles bump into each other, then they must become entangled to a certain degree. Probably the case of a collison is more complex that the one I discussed before (the hydrogen molecule) because in the case of the collision there are more variables involved such as momentum, position, etc.
But in either case, there are combinations of the original tensor product of the separate Hilbert spaces that dissapear. How does this happen? Does entropy for the interacting particles change? How? is there a need to assume some dissipative effect?
I realize a discussion of this phenomenom may become very involved. If anybody here has seen an article on the web where this subject is explained, I would appreciate your pointing us in the right direction. Just remember, we already know what entanglement is, what we want to know is how two particles can become entangled in the first place. We want all the details about what happens when we bring them together. I think an understanding of this is crucial to tackle things such as environment-induced decoherence, the measurement problem, etc.
Once again, If you are knoledgeable about these topics I'll appreciate your guidance.
-ALex Pascual-

11. Nov 27, 2004

### TheDonk

alexepascual is right. The answers haven't been exactly what I was asking for. The #4 post, by jvangael was getting closer, but it is more of the way people can do it instead of the general way it happens.

I'm hoping for an answer like:
When two fundamental particles are within x nanometers away.

I know it probably won't involve distance but there must be a single property that two particles can have that will entangle them.

12. Nov 28, 2004

### masudr

Two particles must have interacted. If they have, then two measurements represented by operators A and B must behave like [A, B] = ih and then we have entanglement.

13. Nov 28, 2004

### blip

What if they each perform a measurement simultaneously, say one vertical and the other horizontal? Afterwords they send their photon through a wave plate of the type corresponding to the measurement they each made.

Would they pass through their respective plates?

14. Nov 28, 2004

### TheDonk

"Two particales must have interacted."
Can you give me an example of how two particles could interact to become entangled? A simple (if possible) step by step process where two particles start off not entangled and become entangled.

I'm not familiar with this equation and I don't know what i and h are. Can you explain what it means without the equation?

15. Nov 28, 2004

### alexepascual

TheDonk:
"i" is the square root of (-1) and "h" is Plank's constant. A and B are what is called "Hermitian Operators" which represent "observables".
If you have little knowledge of quantum mechanics, I suggest you look at some tutorials on the web. Depending on the level at which you want to understand it, the math may become a little intimidating though. In parallel to reading some non-mathematical articles, I suggest you read a book on linear algebra, which is needed for Quantum.
But let me tell you that I think the previous post does not answer your question or mine, and I don't see the conection between what he says and the interaction between two particles.
-Alex-

16. Nov 29, 2004

### Eye_in_the_Sky

First of all let's explain the idea of quantum "entanglement".

Suppose that we have two particles, 1 and 2, with corresponding Hilbert spaces H1 and H2. Suppose that particle 1 is in the state |ψ> Є H1, and that particle 2 is in the state |φ> Є H2, and all of this is before the two particles interact. Then, prior to the interaction, the state of the joint system is simply

|ψ>|φ>.

Now, suppose that the interaction between these two particles is such that

|ψ>|φ> → Σk akk>|φk> ,

where each ak ≠ 0, and there are at least two distinct values for k (and, of course, the |ψk> (|φk>) are linearly independent).

Then, the state of the joint system after the interaction can no longer be written as a simple (tensor) product of one element from H1 with one element from H2 – it must be written as a linear combination of such products. The two particles are now said to be in an "entangled" state.

Next, you ask regarding the interaction itself, referring to it as a sort "bumping" between the two particles:
It sounds like the type of interaction you have in mind is that of a "collision-like" scenario. So, let's use the example of an "elastic collision". Then, with regards to the "bump" itself, there is nothing really special about it. What is special here is that we are dealing with quantum states.

First let's conceptualize the situation classically. Think of two particles (which repel one another) on a collision course as viewed in their center-of-mass frame. If the particles are directed perfectly "head-on", each one will bounce back in exactly the opposite direction. On the other hand, if their lines of flight are slightly "off-center", each one will be deflected by some angle from its original line of flight, such that:

The smaller the distance between the two lines of flight, the greater the angle of deflection.

However, no matter what the distance between the two lines of flight happens to be, we know that:

The momentum of each particle must be equal and opposite to that of the other.

Clearly, the momenta of the two particles are "correlated" ... and this is due to conservation of momentum.

Classically, we have no difficulty conceptualizing the situation. But, quantum mechanically, we find a bit of a 'twist'.

Suppose that the initial wavefunction for each particle has a very sharp momentum, with particle 1 traveling (to a very good approximation) in the +z direction, and particle 2 traveling (to a very good approximation) in the -z direction. Then, in particular, the wavefunction for each particle's position will show a 'spread' in the xy-plane.

Next, after the particles have gone "bump" and have flown well apart, the wavefunction of the joint system will involve a superposition of the various angles of deflection resulting from each of the possible distances between the two lines of flight consistent with the spread of each particle's initial wavefunction in the xy-plane.

Heuristically, looking at a "reduced" wavefunction |θ>, with θ denoting the angle of the line of flight relative to the z-axis, the above interaction can be summarized as:

|0>|π> → ∫a(θ)|θ>|π+θ> dθ .

Thus, there is nothing really 'special' about the "bump" itself. What is 'special' in all of this is that objects which go "bump" are described by quantum states.

Last edited: Nov 29, 2004
17. Nov 29, 2004

### alexepascual

Eye:
Thanks for your detailed answer. I understood most of it, but I still have a lot of questions. My knowledge about the description of multiple-particle systems is kind of defficient. I have a rough grasp of it but there are a lot of things I don't know.
About the original question by TheDonk, he framed the question using the example of a collision between two elementary particles. It may be that he did so because he read some place that particles become entangled after they "bump" into each other. Your analysis of the collision is very detailed and clear. As a matter of fact in your explanation there is a mixture of mathematical expressions with a more intuitive description. I doubt that TheDonk has the mathematical knowledge needed to understand the mathematical description. (I have not asked him how much math or QM he knows), but I am sure he must have learned something from your non-mathematical description, (provided he understands superpositions).
I suspect though, that a collision of two particles may not be the simplest example to study how two particles can become entangled. Wouldn't spin entanglement be easier to understand?
Going back to my difficulties,
I understand that the description of two non-interacting particles that are not entangled whoud consist of a combination of pairs of base states from each separate Hilbert space, and that this is represented by the tensor product of both Hilbert spaces. Now, I have always thought of the tensor product in terms of the combination of base vectors, but I have never thought about what happens to the states of two particles when we describe them in the combined Hilbert space. Assuming non-interaction, what would happen with the complex coefficients? do we just multiply them together for each individual element of the tensor? If so, what about the time dependence?. As I am writing this, I am thinking that it would not make sense to multiply the coefficients because then you might get interferences where there are none. On the other hand, I can see that the probability of finding the composite system in any of the combinations is proportional to the product of the probabilities of each individual base state.
I think what I said a few lines above might be somewhat confising. I said "what happens to the states...". I understand that the states of the individual particles remain unperturbed. What I meant is that I don't understand how to use the coefficients form the individual base states to construct the compound state.
I am now asking you just these simple questions because I think it may be better to tackle one small point at a time.(If I don't understand the simple things I won't be able to undestand the more complex ones). I recall from previous threads that you were very patient with me and I am very grateful for that.

TheDonk:
Eye-in-the_Sky is a very knowledgeable guy and I think he can help us. If you tell us a little more about how much math and quantum mechanics you know, maybe we can give you explanations that not too elementary or too advanced for your level. I consider myself a beginner, but there may be things which I have already understood that I may be able to explain. About myself, although I got a bachellor's degree in physics, which involves two quarters of quantum mechanics and two quarters of classical mechanics, not to mention mathematical methods, I feel that what I studied for school was always in a rush and I never got to understand each of the topics completely. I continue to read books and articles so that I can learn what for one reason or another I didn't learn in school. And I have found this forum to be a great place to learn and to help others.
With respect to the topic of entanglement, I am very optimistic that Eye_in_the_Sky will be able to help us gain a better understanding.

18. Nov 29, 2004

### TheDonk

Thanks to both Eye_In_The_Sky and to alexepascual. I have a relatively good understanding of math and bad in QM. I've taken a course on linear algebra, and I understand at least the basics of vector calculus. That's about the extent of my math knowledge. As for QM I've only heard the hype. I've seen the "possibilities" on tv and I've read some stuff. Unfortunately I don't know what superposition or tensors are but I'm going to relook into them because I tried to understand them a couple years ago.

Eye_in_the_Sky, your explanation helped, tho I'm still confused.

So certain properties of two particles become entangled? To properly explain 2 entangled particles, it isn't enough to just say they are entangled but you would need to say which properties are entangled. Is this right? What are all the properties that can be entangled? Is there anything else needed to explain how to particles are entangled? I guess these questions are off topic...
Maybe someone should explain how the entanglement of the spin of two particles would happen unless it's very similar to the collision explanation. What properties are entangled from the collision entanglement?

19. Nov 29, 2004

### alexepascual

I think it would be correct to say that some property (observable) of the particle is entangled with that same property or another of the other particle, but maybe Eye can correct me.
In the case of the collision, position and momentum become entangled. In Eye's example, if you find one particle to have certain momentum, the momentum of the other particle has to be per force the opposite. You could also draw conclusions about the other particle's position.
In the case of spins, if you have two hydrogen atoms, each has only one electron, and when you put them in a magnetic field, the spin will take an orientation up or down (with respect to the vertical magnetic field).To be more precise, the spin of each atom will be both up and down at the same time (which is one of the main characteristics of quantum systems). This is called a superposition of states.
If you bring both atoms together, they'll form a molecule. But for those alternatives in which both spins are up or both spins are down, it will be impossible to form a molecule because of Pauli's exclusion principle. So, I guess if you were shooting these atoms towards each other, some would "stick" and some would bounce . About the ones that bounce you can say that their spin is mostly pointing in the same direction while you can be sure that those that stuck to form a molecule have their spins pointing in opposite directions. This relationship between the spins is what entanglement is. Now, you can pull the atoms apart and that relationship (if one is up the other is down) will persist, even if the atoms are taken appart a long distance. When I say "one is up, the other one is down" don't take me wrong. Actually they are both in a superposition of "up" and "down". It is just that when you measure spin on one, at that point you collapse the wave function and one of the two states "up" or "down" becomes reality. If you find one atom to be "up" the other one will be down. In all this discussion we are considering only the spin of the electron in each atom. There are other kinds of spin for the atom but they can be ignored. (at least for a simple treatment?). The nice thing about spin states is that they have only two values (up or down). Momentum and position though are continuous variables, which makes the analysis much more complicated.
I said that a particle can be in different states at the same time which are "superposed". That superposition is represented in a vector space where each dimension represents each of the "base states" that make up the superposition. This vector space is called a Hilbert space. The compuond state is represented by a vector in Hilbert state, and how much of each base state goes into the superposition is represented by the magnitude of the "components" of the vector. (the projection of the vector on each of the base vectors). A Hilbert space can have infinite dimmensions, which is always the case for continuous variables.
I suggest again that you look (maybe google?) for some tutorials on quantum mechanics. You need to learn these things if you want to understand entanglement in some depth. The concept of Hilbert space (also called "state space" is easier to understand when considering spin, because for one electron or proton, it is just a two-dimensional space.
As you are working with vectors, you use matrices to manipulate these vectors, and that's where you need to use linear algebra. You should also try to learn the "Dirac notation". It is not too hard and it is fun. You may be able to find some tutorial on it. Oh! now that I remember, "Wikipedia" has a lot of info on quantum mechanics. Do a search in google for Quantum + Wikipedia and I think you'll find it.

20. Nov 29, 2004

### Mike2

Once they are entangled, can you force the spin of one electron to be up so that the distant electron's spin must be down? That would create instant communication, right?