  Quantum information provides fundamentally different computational resources
than classical information. We prove that there is no unitary protocol able to
add unknown quantum states belonging to different Hilbert spaces. This is an
inherent restriction of quantum physics that is related to the impossibility of
copying an arbitrary quantum state, i.e., the no-cloning theorem. Moreover, we
demonstrate that a quantum adder, in absence of an ancillary system, is also
forbidden for a known orthonormal basis. This allows us to propose an
approximate quantum adder that could be implemented in the lab. Finally, we
discuss the distinct character of the forbidden quantum adder for quantum
states and the allowed quantum adder for density matrices.
