  We study overlaps between two regularized boundary states in conformal field
theories. Regularized boundary states are dual to end of the world branes in an
AdS black hole via the AdS/BCFT. Thus they can be regarded as microstates of a
single sided black hole. Owing to the open-closed duality, such an overlap
between two different regularized boundary states is exponentially suppressed
as $\langle \psi_{a} | \psi_{b} \rangle \sim e^{-O(h^{(min)}_{ab})}$, where
$h^{(min)}_{ab}$ is the lowest energy of open strings which connect two
different boundaries $a$ and $b$. Our gravity dual analysis leads to
$h^{(min)}_{ab} = c/24$ for a pure AdS$_3$ gravity. This shows that a
holographic boundary state is a random vector among all left-right symmetric
states, whose number is given by a square root of the number of all black hole
microstates. We also perform a similar computation in higher dimensions, and
find that $h^{( min)}_{ab}$ depends on the tensions of the branes. In our
analysis of holographic boundary states, the off diagonal elements of the inner
products can be computed directly from on-shell gravity actions, as opposed to
earlier calculations of inner products of microstates in two dimensional
gravity.
