Information-Theoretic Aspects of Quantum Copying

Abstract

An information-theoretic approach to quantum copying is discussed, relying on the notion of quantum loss, a quantity that reflects the transmission quality in a noisy quantum channel. More specifically, an entropic nocloning inequality is derived for a Hilbert space of arbitrary dimension, which describes the tradeoff between the losses of the channels leading to the two copies. Then, focusing on quantum bits, a family of Pauli cloning machines is introduced. These machines produce two imperfect copies of a single quantum bit that emerge from two distinct Pauli channels. The balance between the quality of the two copies is shown to result from a genuine complementarity principle. In the special case where the two outputs are associated with depolarizing channels of probability p and p, the domain in (⇏p⇏p)-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. Finally, the class of symmetric Pauli cloning machines is used to provide an upper bound on the quantum capacity of the Pauli channel of probabilities px, py and pz. The capacity is proven to be vanishing if (⇏px⇏py⇏pz) lies outside an ellipsoid whose pole coincides with the depolarizing channel that underlies the universal cloning machine.