<title>A<sc>bstract</sc>
</title>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 <inline-formula><alternatives><tex-math>$$ \left\langle \left.{\psi}_a\right|{\psi}_b\right\rangle \sim {e}^{-O\left({h}_{ab}^{\left(\min \right)}\right)} $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mfenced>
<mml:mrow>
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<mml:msub>
<mml:mi>ψ</mml:mi>
<mml:mi>a</mml:mi>
</mml:msub>
</mml:mfenced>
<mml:msub>
<mml:mi>ψ</mml:mi>
<mml:mi>b</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
<mml:mo>∼</mml:mo>
<mml:msup>
<mml:mi>e</mml:mi>
<mml:mrow>
<mml:mo>−</mml:mo>
<mml:mi>O</mml:mi>
<mml:mfenced>
<mml:msubsup>
<mml:mi>h</mml:mi>
<mml:mi>ab</mml:mi>
<mml:mfenced>
<mml:mo>min</mml:mo>
</mml:mfenced>
</mml:msubsup>
</mml:mfenced>
</mml:mrow>
</mml:msup>
</mml:math></alternatives></inline-formula>, where <inline-formula><alternatives><tex-math>$$ {h}_{ab}^{\left(\min \right)} $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msubsup>
<mml:mi>h</mml:mi>
<mml:mi>ab</mml:mi>
<mml:mfenced>
<mml:mo>min</mml:mo>
</mml:mfenced>
</mml:msubsup>
</mml:math></alternatives></inline-formula> is the lowest energy of open strings which connect two different boundaries <italic>a</italic> and <italic>b</italic>. Our gravity dual analysis leads to <inline-formula><alternatives><tex-math>$$ {h}_{ab}^{\left(\min \right)} $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msubsup>
<mml:mi>h</mml:mi>
<mml:mi>ab</mml:mi>
<mml:mfenced>
<mml:mo>min</mml:mo>
</mml:mfenced>
</mml:msubsup>
</mml:math></alternatives></inline-formula> = <italic>c/</italic>24 for a pure AdS₃ 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 <inline-formula><alternatives><tex-math>$$ {h}_{ab}^{\left(\min \right)} $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msubsup>
<mml:mi>h</mml:mi>
<mml:mi>ab</mml:mi>
<mml:mfenced>
<mml:mo>min</mml:mo>
</mml:mfenced>
</mml:msubsup>
</mml:math></alternatives></inline-formula> 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.