Horacio Casini, Marina Huerta
The path integral representation for the Renyi entanglement entropies of integer index n implies these information measures define operator correlation functions in QFT. We analyze the question of whether the limit n->1, corresponding to the entanglement entropy, can also be represented in terms of a path integral with insertions on the region's boundary, at first order in (n-1). This conjecture has been used in the literature in several occasions, and specially in an attempt to prove the Ryu-Takayanagi holographic entanglement entropy formula. We show it leads to the infinite divisibility of the relevant correlation matrices in the n->1 limit, which is equivalent to an infinite series of polynomial inequalities for the entropies in QFT and the areas of minimal surfaces representing the entanglement entropy in the AdS-CFT context. We check these inequalities in several examples. No counterexample is found in the few known exact results for the entanglement entropy in QFT, but we find counterexamples in the minimal areas corresponding to more complicated geometries. We develop some analytic tools to test the inequalities, and as a byproduct, we show that positivity for the correlation functions is a local property when supplemented with analyticity. We also discuss general situations where infinite divisibility for correlation matrices appears in quantum field theory. This is the case of some exponential operators in the semiclassical and large N limits. This result has analogies to the central limit theorem in probability theory.
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http://arxiv.org/abs/1203.4007
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