Virtual phase-covariant quantum broadcasting for qubits
Reiji Okada, Francesco Buscemi
TL;DR
This work investigates virtual phase-covariant quantum broadcasting for qubits by relaxing unitary covariance to phase-covariance and adding flip covariance, permutation invariance, and classical consistency. The authors fully constrain the structure of the virtual map within this symmetry class, identify a unique map that minimizes the simulation cost, and show that both the cost and the distance to the closest CPTP map are smaller than in the unitary-covariant setting. They find that the closest physical map is the phase-covariant cloning channel, mirroring the relationship seen in unitary covariant broadcasting, and quantify this proximity with a diamond norm distance of $\frac{2}{3}$. Despite these improvements, the virtual phase-covariant broadcasting map remains sample-inefficient and operationally impractical. The work also discusses extensions to qudits and open questions about achieving sample-efficient virtual broadcasting and redistribution rates.
Abstract
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.