title: The Forbidden Quantum Adder.
abstract: 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.
authors.affiliation: Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain.
authors.forename: U
authors.initials: U
authors.lastname: Alvarez-Rodriguez
authors.affiliation: Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain.
authors.forename: M
authors.initials: M
authors.lastname: Sanz
authors.affiliation: Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain.
authors.forename: L
authors.initials: L
authors.lastname: Lamata
authors.affiliation: 1] Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain [2] IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, E- 48013 Bilbao, Spain.
authors.forename: E
authors.initials: E
authors.lastname: Solano