  We study the entanglement entropy, $S_C$, of a massless free scalar field on
the outside region $C$ of two black holes $A$ and $B$ whose radii are $R_1$ and
$R_2$ and how it depends on the distance, $r(\gg R_1,R_2)$, between two black
holes. If we can consider the entanglement entropy as thermodynamic entropy, we
can see the entropic force acting on the two black holes from the $r$
dependence of $S_C$. We develop the computational method based on that of
Bombelli et al to obtain the $r$ dependence of $S_C$ of scalar fields whose
Lagrangian is quadratic with respect to the scalar fields. First we study $S_C$
in $d+1$ dimensional Minkowski spacetime. In this case the state of the
massless free scalar field is the Minkowski vacuum state and we replace two
black holes by two imaginary spheres, and we take the trace over the degrees of
freedom residing in the imaginary spheres. We obtain the leading term of $S_C$
with respect to $1/r$. The result is $S_C=S_A+S_B+\tfrac{1}{r^{2d-2}}
G(R_1,R_2)$, where $S_A$ and $S_B$ are the entanglement entropy on the inside
region of $A$ and $B$, and $G(R_1,R_2) \leq 0$. We do not calculate
$G(R_1,R_2)$ in detail, but we show how to calculate it. In the black hole case
we use the method used in the Minkowski spacetime case with some modifications.
We show that $S_C$ can be expected to be the same form as that in the Minkowski
spacetime case. But in the black hole case, $S_A$ and $S_B$ depend on $r$, so
we do not fully obtain the $r$ dependence of $S_C$. Finally we assume that the
entanglement entropy can be regarded as thermodynamic entropy, and consider the
entropic force acting on two black holes. We argue how to separate the
entanglement entropic force from other force and how to cancel $S_A$ and $S_B$
whose $r$ dependence are not obtained. Then we obtain the physical prediction
which can be tested experimentally in principle.
