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In this note we investigate what is the best L^p-norm in order to describethe relation between the evolution of the state of a bilinear quantum systemwith the L^p-norm of the external field. Although L^2 has a structure more easyto handle, the L^1 norm is more suitable for this purpose. Indeed for everyp>1, it is possible to steer, with arbitrary precision, a generic bilinearquantum system from any eigenstate of the free Hamiltonian to any other with acontrol of arbitrary small L^p norm. Explicit optimal costs for the L^1 normare computed on an example.

Which notion of energy for bilinear quantum systems?