It is not uncommon to say that the non-locality of quantum mechanics is equivalent to the following computer analogy: if you are trying to model an entangled two spin system, then even if the spins are simulated to move apart from each other, you need to do the calculations in the same computer, that is, you cannot take one computer for each spin and separate them to do the calculations, the two computers have to remain together. I heard Leonard Susskind saying this in one of his classes, and there is also the quantum randi challenge https://arxiv.org/abs/1207.5294, which is used to debunk local theories of quantum mechanics.
My question is: can this analogy be formally proved from quantum mechanics, is it so obvious that it does not need any proof, or is it not even correct?
Best Answer
In quantum mechanics, possible states of a system are described by a Hilbert space. If you have two systems, with state spaces $\mathcal{H}_1, \mathcal{H}_2$ then the combined system is described by $\mathcal{H}_1 \otimes \mathcal{H}_2$, which consists of linear combinations of product states like $|\phi \rangle \otimes |\psi \rangle$. So e.g. for a pair of qubits you can have states like
$$\frac{1}{\sqrt{2}} \left( |0\rangle \otimes |0\rangle + |1\rangle \otimes |1\rangle \right)$$
which cannot be written as $|\phi\rangle \otimes |\psi\rangle$ for any choice of $|\phi\rangle, |\psi\rangle$. These states are called entangled.
The point is you cannot describe this state as putting together a state for the first qubit and a state for the second qubit. It only makes sense as a state for both at the same time. This is what Susskind means: you can't "store the state of the first qubit" in one computer and "store the state of the second qubit" in another computer, because they don't even have individual states: they are entangled.
This poses a challenge for thinking about locality in quantum mechanics, for what does locality even mean when your very description of nature is non-local? Nevertheless, 'operational' locality can be rescued, in the sense of the No Communication theorem for instance -- it is not possible to use entanglement to send messages faster than the speed of light.