Part probabilistic cloning of linearly dependent states

TL;DR

This paper investigates the conditions under which linearly dependent quantum states can be probabilistically cloned, identifying specific criteria for subsets of states and exploring optimal cloning efficiencies.

Contribution

It extends probabilistic quantum cloning theory to linearly dependent states, establishing conditions for cloneability and analyzing maximum efficiencies.

Findings

A linearly independent subset can be cloned if no state is a superposition of others.

Cloning efficiency depends on the linear relations among states.

Optimal efficiencies are characterized for certain state sets.

Abstract

From Ref. [Phys. Rev. Lett. 80(1998)4999] one knows that the quantum states secretly chosen from a certain set can be probabilistically cloned with positive cloning efficiencies if and only if all the states in the set are linearly independent. In this paper, we focus on the probabilistic quantum cloning (PQC) of linearly dependent states with nonnegative cloning efficiencies. We show that a linearly independent subset of the linearly dependent quantum states can be probabilistically cloned if and only if any one state in the subset can not be expressed as the linear superposition of the other states in the set. The optimal possible cloning efficiencies are also investigated.