On optimal cloning and incompatibility

Abstract

We investigate the role of symmetric quantum cloning machines (QCMs) in quantifying the mutual incompatibility of quantum observables. Specifically, we identify a cloning-based incompatibility measure whereby the incompatibility of a set of observables maybe quantified in terms of how well a uniform ensemble of their eigenstates can be cloned via a symmetric QCM. We show that this new incompatibility measure is faithful since it vanishes only for commuting observables. We prove an upper bound for for any set of observables in a finite-dimensional system and show that the upper bound is attained if and only if the observables are mutually unbiased. Finally, we use our formalism to obtain the optimal quantum cloner for a pair of qubit observables. Our work marks an important step in formalising the connection between two fundamental concepts in quantum information theory, namely, the no-cloning principle and the existence of incompatible observables in quantum theory.

Measuring Information Leakage in Non-stochastic Brute-Force Guessing

“…Remark 2 (Incompatibility) A similar measure to L α ({ρ x } x∈X ) in Proposition 6 has been proposed as a measure of mutual incompatibility of states within {ρ x } x∈X , albeit for pure states and based on symmetric approximate cloning [19]. Here, L α ({ρ x } x∈X ) generalizes this notion of mutual incompatibility to asymmetric approximate cloning and potentially mixed states.…”

“…We prove that global depolarizing noise, an important family of physical noise in quantum devices [16], reduces the gentle quantum leakage and therefore, can be used as a mechanism for mitigating security and privacy threats, albeit at the risk of sacrificing utility [17], [18]. We also provide a lower bound for gentle quantum leakage based on asymmetric approximate cloning [19]. A numerical example, based on the encoding in the BB84 [1], is used to demonstrate the results.…”