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A simple argument shows that negative energy cannot be isolated far away from positive energy in a conformal field theory and strongly constrains its possible dispersal.
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Without relating it to relative entropy, this has been discussed in the context of the algebraic formulation of QFT. See for example H.J. Borchers, “On the use of modular groups in quantum field theory,” Ann. Inst. H. Poincaré 64, 331-382 (1996)
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A. Borde, L. H. Ford and T. A. Roman, “Constraints on spatial distributions of negative energy,” Phys. Rev. D 65
2002
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C. J. Fewster, “Lectures on quantum energy inequalities,” arXiv:1208.5399 [gr-qc]
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2012
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