Why is quantum teleportation needed to transmit a state? Suppose that Alice wants to teleport the state $|\psi \rangle = \alpha |0 \rangle + \beta |1 \rangle$ to Bob. My issue is why quantum teleportation is needed to transmit this state. Why is it not possible for Alice to do many measurements which could give information on the coefficients and then transmit this information classically to Bob?
 A: If all you want to do is recreate a single quantum state somewhere else, and you have the ability to create exactly this state as many times as you want, and you don't care about exact recreation fidelity, and both sites have the same capability to generate an arbitrary state, then yes, you could do that entirely with classical information.
But if one of those conditions isn't fulfilled, then quantum teleportation becomes useful. If you only have one copy of this state (suppose, for example, it was generated by some extremely rare interaction), you can't perform multiple measurements on it without destroying it (since a measurement necessarily alters the state). If you really need exact fidelity at the receiving end, then quantum teleportation allows that (at least in principle), while any measurement is intrinsically noisy to some degree. If the receiving end is limited in its capability to produce states, then the only way for them to receive an arbitrary state is to quantum-teleport it.
And finally, quantum teleportation doesn't just exist for its own sake. It's used in other applications, like tamper-evident cryptographic key exchange. And in those applications, replacing it with classical information really isn't possible.
These are just a few possibilities where it could be useful; I don't claim it's anywhere near an exhaustive list.
A: Just to add, if you want to decribe a quantum state completely, you have to employ so-called quantum tomography. In simple words, you have to measure the quantum state from different angles of view (i.e. in different basis - computational, Hadamard, circular). After that, you have to compute density matrix and find its eigenstates.
This is a very crude description of the tomography, have a look here for more information.
The procedure described has two main drawback:

*

*You have to have many copies of measured states

*Complexity of the tomography is exponential in number of qubits involved

In comparison, quantum teleportation is easier.
