  We derive several bounds on fidelity between quantum states. In particular we
show that fidelity is bounded from above by a simple to compute quantity we
call super--fidelity. It is analogous to another quantity called sub--fidelity.
For any two states of a two--dimensional quantum system (N=2) all three
quantities coincide. We demonstrate that sub-- and super--fidelity are concave
functions. We also show that super--fidelity is super--multiplicative while
sub--fidelity is sub--multiplicative and design feasible schemes to measure
these quantities in an experiment. Super--fidelity can be used to define a
distance between quantum states. With respect to this metric the set of quantum
states forms a part of a $N^2-1$ dimensional hypersphere.
