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In quantum mechanics, the state of a system is described by a mathematical entity called the wave function or state vector. (A system can mean basically any physical object, but think of particles such as photons, atoms, ions or molecules, maybe more than just one.)
According to Heisenberg's uncertainty principle, it is impossible even in principle to completely measure the state vector of an idividual quantum system, as its different constituents ("amplitudes") are complementary: If you measure one quantity to precisely obtain its value, you necessarily disturb (change) some other quantity so that you cannot find out its value. Hence, it is impossible to get all the "quantum information", which completely describes a system, out of it.
This makes it impossible to create an exact duplicate of it. The no-cloning theorem asserts that this still holds if you try to find other techniques than measuring and recreating the system.
Surprisingly, you can however transfer the complete state of a system onto a similar system which is at some other place. The original system loses its state, of course, in order to follow the no-cloning postulate.
Let's say that Alice has a rubidium atom (the element physicists in this field like to use for their experiments), which is in its ground stateIn physics, the ground state of a quantum mechanical system is its lowest- energy state. An excited state is any state with energy greater than the ground state. If more than one ground state exists, they are said to be degenerate''. Many systems have deg, and Bob has also has such an atom, as well in its ground state. It is important to see, that these two atoms are indistinguishableIdentical particles or indistinguishable particles are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, as well as composite microscopic particles su, that means, that there really is no difference between them.
If Alice and Bob had, say, two glass balls, which exactly look alike, and they exchanged them, then something would change. If you had a powerful microscope, you could certainly find some difference between the two balls. For atoms of the same kind and in the same quantum state, however, there really is no difference at all. The physical situation with Alice having the first atom and Bob the second is exactly the same as vice versa. 1 In a certain sense, it is even wrong to say that the two atoms have any individuality or identity. It would be more appropriate to say that the two locations in space both have the property that the fundamental quantum fields have those values which define the ground state of the rubidium atom.
Now, imagine Alice's atom being in some complicated (excited) quantum state. Assume that we do not know this quantum state -- and of course, we cannot find out by inspection (measurement). But what we can do is to teleport the quantum state to Bob's rubidium atom. After this operation, Bob's atom is exactly in the state that Alice's atom was before.
Now note that Bob's atom afterwards is indistinguishable from the Alice's atom before. In a way, the two are the same -- because it does not make sense to claim that two atoms are different only because they are at different locations. If Alice had gone to Bob and given him the atom we would have exactly the same physical situation.
But Alice and Bob were not required to meet. They only needed to share entanglement.
To see what this means, let us abstract to qubitA qubit is not to be confused with a cubit, which is an ancient measure of length. A qubit ( quantum + bit; pronounced /kyoobit/ [1 ) is a unit of quantum information. That information is described by state in a 2-level quantum mechanical system, whose tws. The atoms are now in states of the form α |ground state> + β |first excited state> (using bra-ket notationBra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract vectors and linear functionals in pure mathematics. It is so called because the inner product of two states).
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