  In this paper we introduce a hyperbolic distance $\delta$ on the
noncommutative open ball $[B(H)^n]_1$, where $B(H)$ is the algebra of all
bounded linear operators on a Hilbert space $H$, which is a noncommutative
extension of the Poincare-Bergman metric on the open unit ball of $C^n$. We
prove that $\delta$ is invariant under the action of the group
$Aut([B(H)^n]_1)$ of all free holomorphic automorphisms of $[B(\cH)^n]_1$, and
show that the $\delta$-topology and the usual operator norm topology coincide
on $[B(H)^n]_1$. Moreover, we prove that $[B(H)^n]_1$ is a complete metric
space with respect to the hyperbolic metric and obtained an explicit formula
for $\delta$ in terms of the reconstruction operator. A Schwarz-Pick lemma for
bounded free holomorphic functions on $[B(H)^n]_1$, with respect to the
hyperbolic metric, is also obtained.
