The no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state. How does that fit together with Quantum tomography, a process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states? It seems that if I want to reconstruct a quantum state, e.g. as a result of a quantum algorithm, first I have to make many copies of the (unknown) state, which I then measure. Why is there no contradiction?
Quantum tomography requires many copies of a quantum state, so we typically use it to describe some hardware device. For example, say you have a quantum widget where you can press a button and it will give you a pair of Bell state photons most of the time, but sometimes just gives a classical pair of photons. By pressing that button a bunch and measuring those photon pairs in a number of bases, you can statistically reconstruct density matrix corresponding to the quantum state my widget's output photon pairs.
Now the no-cloning theorem says that given a single copy of an unknown quantum state, you cannot create a perfect copy of it. Going back to the widget, if you press the button once, there's no way of deterministically determining if the pair of photons you got from that one press are in a Bell state or a classical state since you only get one measurement before you perturb your photon pair, and there's no way to copy that state to be able to make more measurements.
TLDR: Quantum tomography lets you statistically describe the outputs of a quantum source, while the no-cloning theorem limits what you learn from a single output of a quantum source.