Minimal sets determining universal and phase-covariant quantum cloning

TL;DR

This paper identifies minimal input sets that fully determine universal and phase-covariant quantum cloning machines, simplifying their testing and advancing understanding of quantum cloning capabilities.

Contribution

It reveals the minimal input states needed to completely specify universal and phase-covariant quantum cloning machines, refining previous knowledge and practical testing methods.

Findings

Universal cloning determined by four tetrahedral states.

Phase-covariant cloning determined by three equatorial states.

Results relate to quantum cryptography state sets.

Abstract

We study the minimal input sets which can determine completely the universal and the phase-covariant quantum cloning machines. We find that the universal quantum cloning machine, which can copy arbitrary input qubit equally well, however can be determined completely by only four input states located at the four vertices of a tetrahedron. The phase-covariant quantum cloning machine, which can copy all qubits located on the equator of the Bloch sphere, can be determined by three equatorial qubits with equal angular distance. These results sharpen further the well-known results that BB84 states and six-states used in quantum cryptography can determine completely the phase-covariant and universal quantum cloning machines. This concludes the study of the power of universal and phase-covariant quantum cloning, i.e., from minimal input sets necessarily to full input sets by definition. This can simplify dramatically the testing of whether the quantum clone machines are successful or not, we only need to check that the minimal input sets can be cloned optimally.