Abstract
We derive the property of strong superadditivity of mutual information arising from the Markov property of the vacuum state in a conformal field theory and strong subadditivity of entanglement entropy. We show this inequality encodes unitarity bounds for different types of fields. These unitarity bounds are precisely the ones that saturate for free fields. This has a natural explanation in terms of the possibility of localizing algebras on null surfaces. A particular continuity property of mutual information characterizes free fields from the entropic point of view. We derive a general formula for the leading long distance term of the mutual information for regions of arbitrary shape which involves the modular flow of these regions. We obtain the general form of this leading term for two spheres with arbitrary orientations in spacetime, and for primary fields of any tensor representation. For free fields we further obtain the explicit form of the leading term for arbitrary regions with boundaries on null cones.
Highlights
In a conformal field theory (CFT), and when the lowest dimension operator is a scalar of dimension ∆, and the separation L between the regions is much larger than their sizes, we have
We find that saturation of superadditivity implies a certain geometric continuity of the mutual information that can only hold for free fields
As a generalized free field with the relevant two point function of dimension ∆ is a valid CFT, having the operator with dimension ∆ as the lowest dimensional operator, the bounds apply to any field in any CFT, disregarding the presence of other lower dimensional fields
Summary
For a CFT, the entanglement entropy of the vacuum in regions having a boundary on the null cone satisfies the Markov property. This is the saturation of strong subadditivity [14]. Calling A1, A2 to the two regions with boundaries on the null cone, we have for the entropies. Given two arbitrary space-like separated regions A, B we can compute the mutual information. Taking two regions A1, A2 with boundaries on the null cone as above, and another spatially separated region B (see figure 1), we can compute. Valid when one of the entries in the mutual information satisfies the Markov equation (2.1). In the rest of the work we will study implications of this inequality, and its infinitesimal form, for CFTs
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