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Applying the thermo-field double formalism to extremal black holes in AdSwith a macroscopic horizon, we show that (1) there exists a natural basis forthe degenerate microstates of an extremal black hole, and (2) clusterdecomposition in the bulk implies that all correlators are exactly the same forevery microstate of the extremal black hole. The latter statement can beinterpreted in two ways. First, at the fully non-perturbative level of AdS/CFTat finite N, it means that cluster decomposition does not hold in the bulk.This may be viewed as a sharp manifestation of the bulk non-locality at finiteN. Second, at the level of the perturbation theory in 1/N, in which case weexpect the bulk cluster decomposition, no measurement of either boundaryoperators or bulk field operators can distinguish the different microstates.The latter interpretation may exclude some versions of the fuzzball conjecturethat assert that different microstates of a black hole are realized in the bulkas different metric and field configurations.

Bulk Cluster Decomposition in AdS/CFT and A No-Go Theorem for Correlators in Microstates of Extremal Black Holes