A fundamental bound to quantum information processing

Consider a quantum system prepared in an input state which must be driven in a given target state. Focusing on the quantum part of the process and taking as free the cost of the classical part of the transformation, It can be identified a geometric cost function which quantifies the difficulty of the process in terms of how different it is from a classical process. This quantity determines a lower bound to the number of commuting unitary transformations required to complete the task in the discrete case, but I extended the result to the continuos case. I then discuss a geometric bound to the energetic cost of a quantum transformation and its relation to the optimal path in few but relevant examples.