Asymmetric quantum cloning machines in any dimension

Abstract

A family of asymmetric cloning machines for $N$-dimensional quantum states is introduced. These machines produce two imperfect copies of a single state that emerge from two distinct Heisenberg channels. The tradeoff between the quality of these copies is shown to result from a complementarity akin to Heisenberg uncertainty principle. A no-cloning inequality is derived for isotropic cloners: if ${\pi }_a$ and ${\pi }_b$ are the depolarizing fractions associated with the two copies, the domain in $(\sqrt{{\pi }_a},\sqrt{{\pi }_b})$-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. More generally, a no-cloning uncertainty relation is discussed, quantifying the impossibility of copying imposed by quantum mechanics. Finally, an asymmetric Pauli cloning machine is defined that makes two approximate copies of a quantum bit, while the input-to-output operation underlying each copy is a (distinct) Pauli channel. The class of symmetric Pauli cloning machines is shown to provide an upper bound on the quantum capacity of the Pauli channel of probabilities $p_x$, $p_y$ and $p_z$.

18 pages RevTeX, 3 Postscript figures; new discussion on no-cloning uncertainty relations, several corrections, added references

References in corpus (3)

• Optimal Cloning of Pure States
• Quantum cloning without signaling
• Optimal Copying of One Quantum Bit

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