"Broadcasting", namely distributing information over many users, suffers in-principle limitations when the information is quantum. This poses a critical issue in quantum information theory, for distributed processing and networked communications. For pure states ideal broadcasting coincides with the so-called "quantum cloning", describing an hypothetical ideal device capable of producing from a finite number N of copies of a state (drawn from a set) a larger number M>N of output copies of the same state. Since such a transformation is not isometric, it cannot be achieved by any physical machine for a quantum state drawn from a non orthogonal set: this is essentially the content of the "no-cloning" theorem. For mixed states the situation is quite different, since from the point of view of each single user a local marginal mixed state is indistinguishable from the partial trace of an entangled state, and there are infinitely many joint output states that correspond to ideal broadcasting. Indeed, for sufficiently large number
N of input copies, not only ideal broadcasting of noncommuting mixed states is possible, but one can even purify the state in the process. Such state purification with an increasing number of copies has been named "superbroadcasting". In this paper we will review some recent results on superbroadcasting of qubits, for two different sets of input states, corresponding to universally covariant broadcasting and to phase-covariant broadcasting of equatorial states.
We address the problem of broadcasting
N copies of a generic qubit state to
M>N copies by estimating its direction and preparing a suitable output state according to the outcome of the estimate. This semiclassical broadcasting protocol is more restrictive than a general one, since it requires an intermediate step where classical information is extracted and processed. However, we prove that a suboptimal superbroadcasting, namely broadcasting with simultaneous purification of the local output states with respect to the input ones, is possible. We show that in the asymptotic limit of $M\ensuremath{\rightarrow}\ensuremath{\infty}$ the purification rate converges to the optimal one, proving the conjecture that optimal broadcasting and state estimation are asymptotically equivalent. We also show that it is possible to achieve superbroadcasting with simultaneous inversion of the Bloch vector direction (universal NOT). We prove that in this case the semiclassical procedure of state estimation and preparation turns out to be optimal. We finally analyze semiclassical superbroadcasting in the phase-covariant case.