Sub- and super-fidelity as bounds for quantum fidelity

Abstract

We derive several bounds on fidelity between quantum states. In particular we show thatfidelity 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 demonstratethat sub- and super-fidelity are concave functions. We also show that super-fidelity issuper-multiplicative while sub-fidelity is sub-multiplicative and design feasible schemesto measure these quantities in an experiment. Super-fidelity can be used to define adistance between quantum states. With respect to this metric the set of quantum statesforms a part of a N2 - 1 dimensional hypersphere.