Fidelity of dynamical maps

We introduce the concept of fidelity for dynamical maps in an open quantum system scenario. We derive an inequality linking this quantity to the distinguishability of the inducing environmental states. Our inequality imposes constraints on the allowed set of dynamical maps arising from the microscopic description of system plus environment. Remarkably, the inequality involves only the states of the environment and the dynamical map of the open system and, therefore, does not rely on the knowledge of either the microscopic interaction Hamiltonian or the environmental Hamiltonian characteristic parameters. We demonstrate the power of our result by applying it to two different scenarios: quantum programming and quantum probing. In the first case we use it to derive bounds on the dimension of the processor for approximate programming of unitaries. In the second case we present an intriguing proof-of-principle demonstration of the ability to extract information on the environment via a quantum probe without any a priori assumption on the form of the system-environment coupling Hamiltonian.