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We present two measures of distance between quantum processes based on the superfidelity, introduced recently to provide an upper bound for quantum fidelity. We show that the introduced measures partially fulfill the requirements for distance measure between quantum processes. We also argue that they can be especially useful as diagnostic measures to get preliminary knowledge about imperfections in an experimental setup. In particular we provide quantum circuit which can be used to measure the superfidelity between quantum processes.

We study operators acting on a composite Hilbert space and investigate their local numerical range, local spectral radius and local $C$--spectral radius. Concrete bounds for the local numerical range for Hermitian operators are derived. Local numerical range of a non-Hermitian operator forms a subset of the standard numerical range. While the latter set is convex, the local range needs not to be convex nor simply connected. Local numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.

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.