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Is it possible to clone physical objects, as a computer does with any file?

For example, can we do a Ctrl + C and Ctrl + V of the quantum state, and by extension, of the atoms of a car?

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No. There is an entire theorem in quantum mechanics which forbids cloning of quantum particles, called the No cloning theorem. And also, you cannot measure each and every property or quantum state of a particle precisely. So, no, perfect cloning is not possible.

If you want a baseline level explanation of this theorem, this Minutephysics video is a nice starting point.

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As @PNS points out, there is a theorem called the No-Cloning Theorem which says that cloning of a quantum state is prohibited by the principles of quantum mechanics. However, it is important to understand the specific definition of "cloning" in this context. Essentially, the theorem says the following:

It is impossible to have a process that reliably produces a state $\vert\phi\rangle\otimes\vert\phi\rangle$ given an unknown arbitrary state $\vert\phi\rangle$.

The usual idea of copying something would be to first read it and then reproduce it. One cannot simply read a quantum state due to the following simple reason: as soon as one measures an observable on the unknown arbitrary state $\vert\phi\rangle$, the measurement would collapse the state into one of the eigenstates of the observable and thus, we would lose all information about the initial state $\vert \phi\rangle$ except that it wasn't orthogonal to the eigenstate of the observable in which we found it to be post-measurement. Another way to copy a quantum state might be to let another quantum state interact with it and become a copy of it without us ever needing to do any measurement. This cannot generically work due to the unitarity of time-evolution in quantum mechanics. Thus, the no-cloning theorem.

Notice that given infinitely many copies of an unknown arbitrary state $|\phi\rangle$, one can read off $\vert\phi\rangle$ by performing measurements corresponding to a tomographically complete set of observables. Once the unknown state becomes known to us, we can simply design a process that prepares the known quantum state.

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Is it possible to clone physical objects, as a computer does with any file?

As three dimensional printers coming out with identical objects show, it can be done within classical physics concepts.

Identical within the measurements of classical physics.

For example, can we do a Ctrl + C and Ctrl + V of the quantum state, and by extension, of the atoms of a car?

As the other answers state, not at the quantum framework, asking for identity of wavefunctions.

BUT crystals and the creation of new crystals from seeds depend on quantum mechanics, and macroscopically can be thought as clones of each other, because it is only very special measurements that would show the probabilistic differences between two perfect crystals macroscopically.

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