In discrete models, such as spin chains, the entanglement between a pair of particles in a chain has been shown to vanish beyond a certain separation. In the continuum, a quantum field ø(x) at a point represents a single degree of freedom, thus at a region of finite size there are infinite separate degrees of freedom. We show that as a consequence, in contrast to discrete models, the ground state of a free, quantized and relativistic field exhibits entanglement between any pair of arbitrarily separated finite regions. We also provide a lower bound on the decay rate of the entanglement as a function of the separation length between the regions and briefly discuss the physical reasons behind this different behaviour of discrete and continuous systems.
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We thank Y. Aharonov, L. Vaidman and S. Popescu for helpful comments, J. I. Cirac for suggesting the analogy with ion chains in a trap, I. Klich for helpful ideas and A. Botero for many invaluable discussions. We acknowledge support from the Israel Science Foundation (Grant No. 62/01-1).