The Forbidden Quantum Adder

TL;DR

This paper proves the fundamental impossibility of creating a universal quantum adder for unknown states, linking it to the no-cloning theorem, and proposes an approximate quantum adder for practical implementation.

Contribution

It establishes the no-go theorem for quantum adders of unknown states and introduces an approximate adder feasible for experimental realization.

Findings

No unitary protocol can add unknown quantum states.

An exact quantum adder is forbidden for orthonormal basis states without an ancilla.

An approximate quantum adder can be implemented in the lab.

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.