  According to the classical Penrose inequality, the mass at spatial infinity
is bounded from below by a function of the area of certain trapped surfaces. We
exhibit quantum field theory states that violate this relation at the
semiclassical level. We formulate a Quantum Penrose Inequality, by replacing
the area with the generalized entropy of the lightsheet of an appropriate
quantum trapped surface. We perform a number of nontrivial tests of our
proposal, and we consider and rule out alternative formulations. We also
discuss the relation to weak cosmic censorhip.
