Entanglement: What is it and how is it measured?

[cybercrypt13]

I've been doing more reading so forgive me. I read all the time where testing is being done with entangled particles to test their spin but no real explanation of how you entangle them.

The best I can surmise is that they are two particles that have been run into each other and bounce off in their own directions but because they've been introduced they somehow link up with each other thereafter.

Could anyone provide me a worldly view of what is meant by this term?

Thanks,

glenn

 

[jostpuur]

One thing that is not often emphasized in quantum mechanics, is that wave functions do not always describe individual particles, but systems of particles.

Suppose you have one particle, and there is a coordinate $$q_1$$ that describes its state classicaly. In quantum mechanics its state is then described by a wavefunction $$\psi_1(q_1)$$. If you then have another particle, and a coordinate $$q_2$$ for it, then in general you cannot describe this system in quantum mechanics with two wave functions $$\psi_1(q_1)$$ and $$\psi_2(q_2)$$, but you need one $$\psi(q_1,q_2)$$, that attaches a complex amplitude for each possible combination of the coordinates.

For one particle the state vector is $$|\psi\rangle=\textrm{sum}_{q_1}\psi_1(q_1)|q_1\rangle$$. For two particle system the basis is then $$|q_1\rangle |q_2\rangle$$, and the state vector is $$|\psi\rangle=\textrm{sum}_{q_1,q_2}\psi(q_1,q_2)|q_1\rangle |q_2\rangle$$.

If the state is for example $$|0\rangle |2\rangle + |1\rangle |3\rangle$$ (these are some hypothetical observabels, not important for the idea), then the particles are entangled. If you measure the $$q_1$$, you get either 0 or 1, but at the same time the entire system collapses to either 0,2 or 1,3 state. So the second particle becomes involved in the collapse. I guess that's the point in engtanglement.

 

[cybercrypt13]

If the state is for example $$|0\rangle |2\rangle + |1\rangle |3\rangle$$ (these are some hypothetical observabels, not important for the idea), then the particles are entangled. If you measure the $$q_1$$, you get either 0 or 1, but at the same time the entire system collapses to either 0,2 or 1,3 state. So the second particle becomes involved in the collapse. I guess that's the point in engtanglement.

I appreciate the answer but guess you missed the part about a worldly explanation. There are many experiments that all refer to entangling two particles of which we can not see without eyes or with any viewing device. It is just stated that they are entangled like you take a piece of string and tie them together.

I've been trying to determine what is going on exactly that you or anyone else can be 100% positive that anything has been entangled and what exactly you do to make this happen. Do we have a machine with an entangle button on it that's twists them together? :-)

I'm studying this QM stuff as hard as I can trying to understand what is going on and honestly, so far its like a bunch of double talk where no one really wants to answer any questions. Maybe your answer is easily understood by you but it makes no sense what so ever to me.

Could anyone just answer the question in a simple way that someone wanting to understand can understand?

Thanks a bunch,

glenn

 

[ueit]

I've been trying to determine what is going on exactly that you or anyone else can be 100% positive that anything has been entangled and what exactly you do to make this happen. Do we have a machine with an entangle button on it that's twists them together? :-)

If a diatomic molecule splits (like Hg2->2Hg*) the two atoms are entangled. Entangled photons are produced in PDC sources (parametric down converter). Google for those words to get more info.

 

[jambaugh]

Entangled particles are not physically connected in anyway. They are just particles created in such a way that their properties are strongly correlated.

To understand this takes some subtile thinking. A quantum particle may for example have only two possible "states" but these two can be resolved in a infinite number (continuum) of ways. It helps to avoid the (classical) idea of the particle "being in a state". For example one speaks of the spin of an electron as being one of "up" or "down" but this is resolving that spin in one specific direction (traditionally z). One may however pick any of the continuum of directions as the new "up"vs "down" and resolve the two outcomes in that direction. Any such choice negates the exactness of an alternative choice. We at best can make probabilistic predictions about observations of spin in a direction other than the one we previously observed the particle (as either up or down).

Now consider a combined system of two "2-state" particles you get 2x2 or four "states". But there are also multiplicatively many more ways to
resolve these four "states" than just those of separately resolving the "states" of each particle. Included are observations of the two particle system which maximally resolve its combined state but give no information about anyone of the particles. A later observation of how one of the particles behaves will not be predictible. Instead one may with certainty predict that this outcome will be say exactly the same, or in another case, exactly opposite of the parallel observation of the second particle.

The particles are then said to be entangled.

To be precise an entangled pair of particles is a system wherein a complete observation/preparation has been made which does not commute with those observables we use to identify which of the two particles we are considering (e.g. their position, which is spin up vs spin down in a given direction for the case of an anti-correlated pair, their momentum, and so on.)

Now consider that for a single particle we may make an initial observation. And if we immediately re-observe the particle we will see the same result. The outcomes are exactly correlated. However if we wait then it may be the case that the original observation no longer correlates. However there will be some observation which does. Example: Sugar water will rotate the polarization of light traveling through it. Thus an initial observation of polarization in one direction will correlate with a later observation in a rotated direction.

Now what happens with entangled particles (usually) is that the initial maximally sharp preparation which may occur locally will not correlate perfectly with any local observable after a period of time.

This isn't too weird. An initial position measurement of a particle will not over time correspond to a later position measurement. This because the momentum has been "scrambled" by the act of measuring position.

In the confusion about EPR experiments many confuse this non-locality of the act of observation which is equivalent to the initial preparation with what they see as a necessary non-local cause and effect.

Regards,
James Baugh

 

[cybercrypt13]

Thanks for the reply and information. I'm still a bit confused with how one goes about setting up this entanglement process to be sure the particles are truly entangled.

Thanks,

glenn

 

[DrChinese]

I'm still a bit confused with how one goes about setting up this entanglement process to be sure the particles are truly entangled.

Entangled particles often exhibit properties which unentangled particles do not. For example, their spin/polarization properties are correlated in ways that would violate the laws of chance. Thus it is possible to calibrate an apparatus by looking for maximal violation of certain correlations. For photon pairs, the experimenter looks for the so-called "perfect correlations" (with detectors set at 0 or 90 degrees apart).

Once you have a stream of entangled particle pairs, you can perform experiments (such as a test of Bell's Inequality, verification of the particle nature of light, etc).

 

[vanesch]

Entanglement of systems is, as others pointed out, a concept that follows from the principles of quantum theory, and the fact that we try to slice up the world in "sub systems". It is impossible to give correctly a "day-by-day" (classical) example or analogy of what entanglement is about, as the concept is completely absent in any classical theory.

A very fundamental principle (I consider it *the* fundamental principle) of quantum theory, which includes all its weirdness and is at the same time the basis for its usual mathematical formulation, is: the superposition principle. It is kind of unheard of, but here it goes:
If a system can be in an observable state A, and it can be in an observable state B, then it can also be in any complex projective superposition of these states, and these physical states are physically distinguishable from A and B themselves.

This is the basis for the Hilbert space formulation: to every physical state corresponds a vector in Hilbert (projective) hilbert space (the projective is there because vectors which only differ by a multiplicative complex number are considered to describe the same physical state), and to a specific observation correspond only a small subset of these states, which form an orthogonal basis of the Hilbert space.

The application to a simple system, such as a point particle in Euclidean space, is rather straight-forward. A point particle in Euclidean space can be observed to be at a point p1 (coordinates x1,y1,z1) or p2 (coordinates x2,y2,z2) or p3 or... every possible point in Euclidean space. So there are (orthogonal) basis states corresponding to each one of these "position states" which are written in Dirac notation: |p1>, |p2>, ... |p3> ...

But, using the superposition principle, *all thinkable complex combinations* of all these states are also (distinct) physical states:
c1 |p1> + c2 |p2> + c3 |p3> +...

As such, starting from the (observable) basis states, the entire Hilbert space is spanned, and filled with distinct physical states. A simple way to write such a specific state is by giving a complex number ci to each possible position state |pi> or, giving a complex number to each point in space. But, giving a complex number to each point in space is nothing else but defining a COMPLEX FUNCTION over space, the so-called "wave function" of the state.

Fine. What now with two point particles (a red and a blue one) ?
Well, the observable states are now, for instance, the TWO positions of the two particles. We take the convention that we now write first the red position, and next, the blue position. So, COUPLES of points are now "observable states". We write them |p1,q1>. We also have, |p1,q2> and |p2,q1> etc... To each COUPLE OF POINTS corresponds now an observable (basis) state. We apply again, the superposition principle, and we now find our most general state:
c11 |p1,q1> + c12 |p1,q2> + c21 |p2,q1> + ...
We hence span the entire Hilbert space of states, which is now the HILBERT SPACE OF A 2-PARTICLE SYSTEM.

Again, we can characterise each state with a complex number for each COUPLE OF POINTS. That's nothing else but the "wave function" psi(p,q).

And here comes the crux:
with most psi(p,q) does NOT correspond a single psi(p) and a single psi(q). In other words, MOST "two-point-particle states" are NOT the juxtaposition of a state of the first particle and the state of the second particle. It is only in extremely special cases, that a general complex function psi(p,q) = f(p) g(q). Most of the time, this is not the case.
This is, as shown, a result of the superposition principle (which is at the basis of the entire framework of quantum theory), and has no equivalent in classical physics. Indeed, a 3-particle state in classical physics would be something like (q1,p1,q2,p2,q3,p3) - an element of the phase space of 3 particles, which is an 18-dimensional manifold. Clearly, to such a state corresponds a specific state of the first particle (namely, q1,p1), a specific state of the second particle (namely q2,p2) and a specific state of the third particle (q3,p3).
The state space of a "multi-system" in classical physics is simply the set product of the state spaces of the subsystems. To each element of the multi-system corresponds a "tuple of states" for each of the sub systems.

Not so in quantum theory. Most of the states of a "multi-system" are not a tuple of states of the subsystems, and this comes about because of the superposition principle.

In the case that a quantum state of a multi-system is NOT one of those very special states which are "factorisable" (psi(p,q) = f(p) g(q)), then we say that the quantum state is an ENTANGLED state of the subsystems.

 

[cybercrypt13]

Ok, I appreciate that lengthy explanation and I think I follow what you are saying. However, what I'm really interested in, is the process one goes through to claim entanglement. My original question was if the two particles are fired at each other and when they come into contact they become entangled, or what exact process is used to entangle them?

Can anyone point me to something I could read to better understand what this process involves and exactly what we believe is going on during this entanglement?

thanks,

glenn

 

[DrChinese]

Ok, I appreciate that lengthy explanation and I think I follow what you are saying. However, what I'm really interested in, is the process one goes through to claim entanglement. My original question was if the two particles are fired at each other and when they come into contact they become entangled, or what exact process is used to entangle them?

Can anyone point me to something I could read to better understand what this process involves and exactly what we believe is going on during this entanglement?

thanks,

glenn

Glenn,

You seem to miss where we are pointing you. All you have to do is READ some of the references for entanglement. I will help you out on this.

The standard method of creating entangled photons is called Parametric Down Conversion, or PDC. You have 1 photon go in (from a laser source) and 2 come out. The 2 that come out are ALWAYS an entangled pair. The 1 going in has a wavelength of X (usually around 400nm), while the 2 coming out have a wavelength of 2X. You will notice that energy is conserved with this (since the photon energy is proportional to 1/X).

To give you a basic idea of the process (numbers are not exact): a laser beam pumps perhaps 100 million photons per second into a PDC crystal. About 10,000 entangled pairs come out. (The remaining photons are ignored; this is easy to accomplish because of directional issues and they can also be filtered due to their wavelength.) The pairs come out in time windows that make it very easy to distinguish them. It is like driving in West Texas at 3 in the morning, there is nothing else around for miles - to make an analogy.

Here is a reference that describes the process in fairly straightforward manner:

Entangled photons, nonlocality and Bell inequalities in the undergraduate laboratory

http://marcus.whitman.edu/~beckmk/QM/grangier/Thorn_ajp.pdf

Both of these describe the PDC setup pretty well. IMPORTANT NOTE: you may ask "why do these special crystals create entangled photon pairs?" There has been a lot of discussion of this issue, and I do not think the answer is well understood. Nonetheless, they are entangled as this is very easy to demonstrate. Further, the symmetry the photon pairs obey is itself well defined. I strongly recommend that you take the time to read and understand these basics before proceeding.

 

[vanesch]

My original question was if the two particles are fired at each other and when they come into contact they become entangled, or what exact process is used to entangle them?

In fact, almost any interaction between two systems will entangle their states, if they started out already in a product state.

Consider a generic hamiltonian: H = H1 + H2 + H12, where H1 acts only on the first system, H2 only on the second, and H12 is the interaction term acting on both.

Most of the time, the unitary time evolution operator that follows from this hamiltonian, U(t1,t2), will be such, that when it acts upon a product state f(q1,t1)g(q2,t1), that it will result in an evolved wavefunction h(q1,q2,t2) = U(t1,t2) [ f(q1) g(q2) ] which cannot be written in a product form. Then the two systems are entangled.

 

[jambaugh]

Ok, I appreciate that lengthy explanation and I think I follow what you are saying. However, what I'm really interested in, is the process one goes through to claim entanglement. My original question was if the two particles are fired at each other and when they come into contact they become entangled, or what exact process is used to entangle them?

Can anyone point me to something I could read to better understand what this process involves and exactly what we believe is going on during this entanglement?

thanks,

glenn

Entanglement is really not a property of the pair of particles themselves per se. There is no "Entanglement observable" by which you can empirically test whether a single instance of two particles is or is not "entangled".

Entanglement is rather a property of the preparation mode for a pair of particles. You see that the device producing the particle pairs is "entangling" the particles when you see that all corresponding pair measurements are exactly correlated no matter which observable pair you choose to measure. To see this you necessarily must execute a large number of experiments.

There is however a means of discussing entanglement in the large scale as an empirical quantity, namely entropy. This however deserves a whole new thread so I'll direct you there. Look for a post in this section entitled Is the Entropy of the Universe Zero?.

Regards,
James Baugh