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Quantum entanglement is known to be the exchange of quantum information between two particles at a distance, while quantum superposition is known to be the uncertainty of a particle (or particles) being in several states at once (which could also involve the exchange of quantum information for a particle that is known to be in several locations simultaneously). I was wondering if all of this was nothing more but the exchange of quantum information between different masses, and if this could clear up all the confusion in terms of how quantum systems connect in this field of science. A clear explanation for how both of these quantum phenomena work, and if they really are connected (the exchange of quantum information?) would be much appreciated.

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    $\begingroup$ All entangled states are special cases of superposed states. But not all superpositions are entangled states. $\endgroup$ – Raskolnikov Nov 22 '14 at 8:15
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To answer your question shortly: No, they are not the same phenomenon.

First of all, it is much easier to think of quantum states as vectors (in something called the Hilbert space, but simply put they obey linearity), and not as particles or waves.

Superposition

Let's start with a single particle qubit (since you're talking about quantum information) which has two eigenstates $|0\rangle$ and $|1\rangle$. Because of the linearity of the Hilbert space, any superposition if these states is also a valid state: $$|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$$

This is called the superposition of eigenstates. Note that there is nothing uncertain about it, unless you are thinking what outcome you get if you measure in 0, 1 basis.

Entanglement

To have entanglement you necessarily need two things (degrees of freedom) that are entangled. They don't have to be distinct particles, but let's say they are. Now let me throw some crazy maths at you and then try to explain what is means.

Let's say the particles A and B are in the states $|0\rangle_A$ and $|0\rangle_B$. They together also form a quantum system, which can be described by the tensor product: $$ |\psi_{AB}\rangle = |0\rangle_A \otimes |0\rangle_B$$ Here the total state of the system can be described by specifying the state of particle A and the state of particle B. This is not and entangled system.

An entangled state is a state of two particles, which cannot be described by saying which state particle A is in and which state particle B is in. For example:

$$ |\psi_{AB}\rangle = |0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B \neq |\psi\rangle_A \otimes |\psi\rangle_B$$

There is no way to write the entangled state as a tensor product of two individual states. Now focus, this is the punchline: In an entangled state, particle A cannot be described independently of the particle B.

Answer to your question

Now you really shouldn't think of entanglement as exchange of information, because it happens instantaneously and it is impossible to transmit information instantaneously. Again, it is also impossible to transmit information instantaneously by collapsing the superposition.

How they are connected is put nicely by Raskolnikov in the comment.

So what is really happening in entanglement, if it is not the exchange of quantum information? You could say the particles are correlated, but it is really more than just a classical correlation. In fact, Bell's theorem predicts and upper bound on the "amount of correlation" two particles can have, under the assumption of local realism.

Entanglement was found to violate this upper bound, which means that quantum theory is either:

a) not a local theory (local)

b) does not describe things and their physical properties (realism) - e.g. is about our knowledge about the world

c) both of the above

We don't really know which answer is the right one, but there has been recent progress.

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First of all the words "exchange of information" are not so good. An entanglement is a CONSTRAINT on two particles or more. For instance the famous PHOTON SINGLET is described by the state

(1) |Ψ> = [!/sqrt(2)] {|u>|u> + |v>|v>}

where u is whatever direction in space that we want to choose, and v is perpendicular to x in the polarization plane (which is the plane perpendicular to the propagation direction). In the products |u>|u> and |v>|v> first comes photon 1, then comes photon 2. (Sometimes there is some difference between them in the wavelength s.t. we can tell which is which.)

The photon singlet is emitted by different elements in what is called cascade, i.e. there is a jump of an electron from an excited level to a lower one with emission of photon, then a fall to a final level with emission of the second photon. In these transitions CONSERVATION LAWS of angular momentum of the electron are obeyed, and because of them we cannot get in (1) the states |u>|v> and |v>|u> together with the states |u>|u> and |v>|v>.

So, you see, THE NATURE imposes constraints, and therefore in a measurement of polarization, if on the photon 1 we got polarization u, the same thing we will get on photon 2. We DON'T KNOW, and there is a wide polemic on whether the two photons exchange some SIGNALS between themselves, or not. I mean, imagine the following experiment: the observer 1 picks arbitrarily, at their whim, a direction in space, let's call it w, and get that the photon 1 is polarized perpendicularly to w. We use to say that in this case the state of the photon 2 COLLAPSES to the state perpendicular to w. But how the nature transmits to the photon 2 the fact that the experimenter 1 chose the direction w and photon 1 gave the perpendicular response, we DON'T KNOW. Experiments we performed with such a small interval between the measurement of photon 1 and that of photon 2, that not even a signal of light has the time to pass from the location of experimenter 1 to the location of the experimenter 2. So, the words "exchange of information between particles" are problematic. Is some type of signals moving between the two places? We don't know.

About superposition, that is something else. We can have superposition of the states of one particle, or of two particles, or more. Look at the equality (1). The state of the two photons is a SUPERPOSITION of the two-particle state |u>|u> with the two-particle state |v>|v>. But I think that you refer to the superposition of the states of one particle. Then consider a beam of particles split by a beam splitter into a transmitted and a reflected wave, and concentrate on the fate of one of the particles. We can never detect it on both paths. But, again we can place on both paths detectors at the SAME DISTANCE from the beam-splitter, s.t. when one particle is detected, not even a pulse of light had time to pass from one detector to the other. So, is it some information PASSED between the two detectors? WE DON'T KNOW THE ANSWER.

I am sorry, my answer is disappointing, but for the moment the quantum community is still IGNORANT in this direction. Good luck !

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