Is it enough to tell: lets take two particles in entanglement state and increase distance between them. Can we be sure, that increasing distance will not kill entanglement state? What will be gravity influence?
The answer to the question is, Yes. The subsidiary questions, however, don't make sense.
The statement that two particles are in an entangled state is, unfortunately, misleading. Particles are entangled if (and only if) their joint probability distribution doesn't factorize into a product over the probability distribution of each particle. So there has to be a notion of the space of states of each particle, the evolution operator that acts on the space of states of each and of an interaction that describes how the two-particle states are defined in terms of the states of each particle.
``Increasing the distance'' between the particles just leaves too many things ambiguous. The main reason is that the distance between the particles in real space isn't directly related to the factorization of their probability distributions-that are defined on their space of states, their phase space.
It can be expected that particles that interact through ``short range'' forces'' won't remain in an entangled state, if they are separated at a distance much greater than the interaction range of the force between them.
On the other hand, what must be kept in mind is that the property that particles can be found in ``entangled states'' at all is due to the fact that they are in equilibrium with a bath of quantum fluctuations-that's the reason entanglement is possible without a ``classical'' interaction between them.
Regarding gravity, its presence complicates matters in that the space of states of a quantum particle can't be uniquely defined, if spacetime is curved.
It has been shown by Zeilinger that photons that are spin entangled maintain their entanglement over considerable distance. The only way such an entanglement could be broken is if the spin value can change through an interaction, and in general, photons tend to either continue unchanged or they get absorbed and destroyed.
For entanglement of physical particles, though, since the entanglement involves application of a conservation law, it follows that if they can interact with something else the entanglement is broken, and the probability o that occurring must increase with distance travelled.
Other particles, besides photons, carry spin and all are ``physical" particles. Entanglement doesn’t violate any conservation laws and interactions don’t, necessarily, affect entanglement.
The typical example is with photons. A state of more than one photons is entangled, because photons are bosons. This means that they are subject to’a non-local interaction thag describes the property that the multiparticle wavefunction is symmetric.
The same holds for Cooper pairs in a superconductor and the atoms of a superfluid.
Entanglement usually is because of a conservation law. However, I argue non-locality has not been demonstrated, but is merely assumed. At5 thiws stage I am assuming that you would not say the a cork bouncing around on ocean waves is subject to non-local forces, although I suppose you could say the wave is non-local in as much as the trough is somewhere different from the crest
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