Quantum Cloning Optimal for Joint Measurements

Perfect cloning of unknown quantum systems is forbidden by the laws of quantummechanics [1]. However, a universal optimal cloning has been proposed [2], which has been proved to be optimal in terms of fidelity [3, 4]. A obvious relevant application of such optimal cloning is eavesdropping in quantum cryptography [5]. However, quantum cloning can be of practical interest as a tool to engineer novel schemes of quantum measurements, in particular for joint measurements of noncommuting observables. Quite unexpectedly, as we will show in the following, the universally covariant cloning is not ideal for this purpose. Here, instead, cloning must be optimized for a reduced covariance group, depending on the desired joint measurement, in order to make the measurement over the cloned copies perfectly equivalent to an optimal joint measurement over the original. In the following we consider: i) the case of spin 1/2, and the use of the 1 → 3 universally covariant cloning to achieve a joint measurement of the spin components; ii) the case of harmonic oscillator, along with a 1 → 2 cloning transformation that is not universally covariant. We show that the POVM obtained in the first case does not lead to a minimumadded-noise measurement. On the contrary, the second way allows one to achieve the ideal joint measurement of conjugated variables.