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arXiv:quant-ph/0602125 · Physical Review A

Universal and phase-covariant superbroadcasting for mixed qubit states

Francesco Buscemi, Giacomo Mauro D’Ariano, Chiara Macchiavello, Paolo Perinotti

Published Oct 12, 2006• 13 citations
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Abstract

We describe a general framework to study covariant symmetric broadcasting maps for mixed qubit states. We explicitly derive the optimal $N\ensuremath{\rightarrow}M$ superbroadcasting maps, achieving optimal purification of the single-site output copy, in both the universal and phase-covariant cases. We also study the bipartite entanglement properties of the superbroadcast states.

DOI
10.1103/physreva.74.042309

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We describe a general framework to study covariant symmetric broadcasting maps for mixed qubit states. We explicitly derive the optimal N\ensuremath→MN\ensuremath{\rightarrow}MN\ensuremath→M superbroadcasting maps, achieving optimal purification of the single-site output copy, in both the universal and phase-covariant cases. We also study the bipartite entanglement properties of the superbroadcast states.
"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 NNN 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 NNN copies of a generic qubit state to M>NM>NM>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.
We consider the problem of broadcasting arbitrary states of radiation modes from N to M > N copies by a map that preserves the average value of the field and optimally reduces the total noise in conjugate variables. For N ⩾ 2 the broadcasting can be achieved perfectly, and for sufficiently noisy input states one can even purify the state while broadcasting—the so-called superbroadcasting . For purification ( i.e. M ⩽ N ), the reduction of noise is independent of M . Similar results are proved for broadcasting with phase-conjugation. All the optimal maps can be implemented by linear optics and linear amplification.
Virtual maps allow the simulation of quantum operations by combining physical processes with classical post-processing. Recent work on virtual unitary covariant broadcasting has shown, however, that such maps remain impractical for observable estimation tasks due to poor sample efficiency. Here we investigate whether relaxing the symmetry requirements can improve operational performance, focusing on virtual phase-covariant quantum broadcasting for qubits. We show that imposing phase-covariance, flip covariance, permutation invariance, and classical consistency fully determines the structure of the broadcasting map. Within this family, we identify the unique map that minimizes the simulation cost, and we prove that both the simulation cost and the distance to the closest CPTP map are strictly smaller than in the unitary covariant setting. We also demonstrate that the closest physical map is the optimal phase-covariant cloning channel, mirroring the relation between unitary covariant broadcasting and universal cloning. Despite these improvements, the resulting virtual broadcasting map remains sample-inefficient and is therefore still operationally impractical.
We consider the problem of broadcasting quantum information encoded in the average value of the field from N to M>N copies of mixed states of radiation modes. We derive the broadcasting map that preserves the complex amplitude, while optimally reducing the noise in conjugate quadratures. We find that from two input copies broadcasting is feasible, with the possibility of simultaneous purification (superbroadcasting). We prove similar results for purification (M<=N) and for phase-conjugate broadcasting.