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arXiv:quant-ph/0509170 · Journal of Physics A Mathematical and General

Cloning of observables

Alessandro Ferraro, Matteo Galbiati, Matteo G A Paris

Published Mar 22, 2006• 11 citations
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Abstract

We introduce the concept of cloning for classes of observables and classify cloning machines for qubit systems according to the number of parameters needed to describe the class under investigation. A no-cloning theorem for observables is derived and the connections between cloning of observables and joint measurements of noncommuting observables are elucidated. Relationships with cloning of states and non-demolition measurements are also analysed.

DOI
10.1088/0305-4470/39/14/l02

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We introduce the concept of cloning for classes of observables and classify cloning machines for qubit systems according to the number of parameters needed to describe the class under investigation. A no-cloning theorem for observables is derived and the connections between cloning of observables and joint measurements of noncommuting observables are elucidated. Relationships with cloning of states and non-demolition measurements are also analysed.
Cloning of observables, unlike standard cloning of states, aims at copying the information encoded in the statistics of a class of observables rather then on quantum states themselves. In such a process the emphasis is on the quantum operation (evolution plus measurement) necessary to retrieve the original information. We analyze, for qubit systems, the cloning of a class generated by two noncommuting observables, elucidating the relationship between such a process and joint measurements. This helps in establishing an optimality criterion for cloning of observables. We see that, even if the cloning machine is designed to act on the whole class generated by two noncommuting observables, the same optimal performances of a joint measurement can be attained. Finally, the connection with state dependent cloning is enlightened.
A joint measurement of two observables is a simultaneous measurement of both quantities upon the same quantum system. When two quantum-mechanical observables do not commute, then a joint measurement of these observables cannot be accomplished directly by projective measurements alone. In this paper we shall discuss the use of quantum cloning to perform a joint measurement of two components of spin associated with a qubit system. We introduce cloning schemes which are optimal with respect to this task. The cloning schemes may be thought to work by cloning two components of spin onto their outputs. We compare the proposed cloning machines to existing cloners.