  We study a doubly-bottomed tetra-quark state $(bb\bar{u}\bar{d})$ with
quantum number $I(J^P)=0(1^+)$, denoted by $T_{bb}$, in lattice QCD with the
Non-Relativistic QCD (NRQCD) quark action for $b$ quarks. Employing
$(2+1)$-flavor gauge configurations at $a \approx 0.09$ {fm} on $32^3\times 64$
lattices, we have extracted the coupled channel potential between
$\bar{B}\bar{B}^*$ and $\bar{B}^* \bar{B}^*$ in the HAL QCD method, which
predicts an existence of a bound $T_{bb}$ below the $\bar{B}\bar{B}^*$
threshold. By extrapolating results at $m_\pi\approx 410,\, 570,\, 700$ {MeV}
to the physical pion mass $m_\pi\approx140$ {MeV}, we obtain a biding energy
with its statistical error as $E_{\rm binding}^{\rm (single)} = 155(17)$ MeV
and $E_{\rm binding}^{\rm (coupled)} = 83(10)$ MeV, where ``coupled" means that
effects due to virtual $\bar{B}^* \bar{B}^*$ states are included through the
coupled channel potential, while only a potential for a single
$\bar{B}\bar{B}^*$ channel is used in the analysis for ``single". A comparison
shows that the effect from virtual $\bar{B}^* \bar{B}^*$ states is quite
sizable to the binding energy of $T_{bb}$. We estimate systematic errors to be
$\pm 20$ MeV at most, which are mainly caused by the NRQCD approximation for
$b$ quarks.
