  We introduce the (logarithmic) bipartite fidelity of a quantum system $A\cup
B$ as the (logarithm of the) overlap between its ground-state wave function and
the ground-state one would obtain if the interactions between two complementary
subsystems $A$ and $B$ were switched off. We argue that it should typically
satisfy an area law in dimension $d>1$. In the case of one-dimensional quantum
critical points (QCP) we find that it admits a universal scaling form $\sim \ln
\ell$, where $\ell$ is the typical size of the smaller subsystem. The prefactor
is proportional to the central charge $c$ and depends on the geometry. We also
argue that this quantity can be useful to locate quantum phase transitions,
allows for a reliable determination of the central charge, and in general
exhibits various properties that are similar to the entanglement entropy. Like
the entanglement entropy, it contains subleading universal terms in the case of
a 2D conformal QCP.
