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An asymptotic theory of cloning of classical state families

Cloning, or approximate cloning, is one of basic operations in quantum information processing. In this paper, we deal with cloning of classical states, or probability distribution in asymptotic setting. We study the quality of the approximate (n,rn)-clone, with n being very large and r being constant. The result turns out to be \parallel N(0,r1)-N(0,1)\paralell_1, where N(μ,Σ) is the Gaussian distribution with mean μ and covariance Σ. Notablly, this value does not depend on the the family of porbability distributions to be cloned. The key of the argument is use of local asymptotic normality: If the curve θ\rightarrow P_{θ} is sufficiently smooth in θ, then, the behavior of P_{θ'}^{\otimes n} where θ'-θ=o(\surd(1/n)), is approximated by Gaussian shift. Using this, we reduce the general case to Gaussian shift model.