When I do think of entangled states I usually do think of 2 or more particle states, which can not be written as products of 1 particle states.
But what, if we have just 1 particle being described by a state in a product space? Suppose we send a charge-neutral spin-$\frac{1}{2}$-particle threw a Stern-Gerlach setup and suppose the initial state of the particle was a superposition of the spin up and down states at some initial height: $(|\uparrow\rangle + |\downarrow\rangle) \otimes |\psi_{initial}(y)\rangle $. Now after going threw the magnetic field we have a superposition of the spin up and diverted upwards state and the spin down and diverted downwards state: $|\uparrow\rangle \otimes |\psi_{up}(y)\rangle + |\downarrow\rangle \otimes |\psi_{down}(y)\rangle$.
This state is an element of the product space of the spin- and position-Hilbert-spaces, but can not be written as a product of states of the factor-spaces. So from my understanding the mathematical situation is basically the same as when looking at a 2 particle entangled spin state, only the concrete spaces are different. Now my question is am I missing some crucial difference between the 2 situations and if not, would one call the described 1 particle state as “entangled”?
Best Answer
A particle can have subsystems that can be independently manipulated such as the position and spin of an electron. These subsystems can be made to interact so that information is copied between them and this can generate entanglement. Experiments have been proposed to test such entanglement and such entanglement can violate the Bell inequalities:
https://arxiv.org/abs/0705.0322
https://arxiv.org/abs/2011.08286
https://arxiv.org/abs/2109.01058
https://arxiv.org/abs/2011.11797
Single particle entanglement has been observed with photons (thanks to Valter Moretti for pointing this out in a comment):
https://arxiv.org/abs/2003.09961