Suppose we have two particles with entangled eigenfunctions. Let's say they are in two different regions of space, with different local Hamiltonians. Now we measure one of them, therefore the other. The function that describe the state of both, collapses to the eigenvector of the meassured state. What happens next? Do they evolve as a whole, still entangled, or entanglement is broken forever and they evolved by the action of each local Hamiltonian?
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$\begingroup$ physics.stackexchange.com/questions/203831/… $\endgroup$– alanfCommented Jul 9, 2022 at 10:41
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$\begingroup$ What happens to one particle cannot affect the other particle. If you perfectly correlate all the known variables of two particles (so called entangled) and send them in different directions, You can measure one and obviously know something about the other, because you just spent the time correlating the two. I could be even more specific if you could tell me exactly how you correlated the two particles and exactly what variable was later measured on the one particle. $\endgroup$– Bill AlseptCommented Jul 9, 2022 at 15:48
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Suppose we have two particles with entangled eigenfunctions. Let's say they are in two different regions of space, with different local Hamiltonians.
It is true that two particles that are exactly described by a specific Hamiltonian are entangled, and also that each individual particle's wave function could be approximated by solutions of local Hamiltonians, but note "approximated", those mathematical constructs are not entangled with each other.
Now we measure one of them, therefore the other.
The "therefore" is true 1) for the overall Hamiltonian solutions 2) and for the specific measurements utilizing conservation laws, not for a complete wavefunction distribution ( remember it is a probabilistic theory one is working with)
The function that describe the state of both, collapses to the eigenvector of the measured state.
One does not measure a state, only specific variables, spin, momentum and such for the individual events.
What happens next?
The measurement disturbs the boundary conditions that decide the wavefunction from the Hamiltonian. New solutions come next, independent for each particle according to the new boundary conditions.
Do they evolve as a whole, still entangled,
in this simple example entanglement stops.
The individual Hamiltonians are an approximation, as stated above.
Of course if one accepts that there could be one Hamiltonian with all wave function solutions for all boundary conditions , there will be an entanglement/ correlation of variables, and it will all depend on the boundary conditions and the solution of complicated overall Hamiltonian. Example this recent article :
Researchers in Germany have demonstrated quantum entanglement of two atoms separated by 33 km (20.5 miles) of fiber optics. This is a record distance for this kind of communication and marks a breakthrough towards a fast and secure quantum internet
