  Applying the thermo-field double formalism to extremal black holes in AdS
with a macroscopic horizon, we show that (1) there exists a natural basis for
the degenerate microstates of an extremal black hole, and (2) cluster
decomposition in the bulk implies that all correlators are exactly the same for
every microstate of the extremal black hole. The latter statement can be
interpreted in two ways. First, at the fully non-perturbative level of AdS/CFT
at 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 finite
N. Second, at the level of the perturbation theory in 1/N, in which case we
expect the bulk cluster decomposition, no measurement of either boundary
operators or bulk field operators can distinguish the different microstates.
The latter interpretation may exclude some versions of the fuzzball conjecture
that assert that different microstates of a black hole are realized in the bulk
as different metric and field configurations.
