Virtual phase-covariant quantum broadcasting for qubits

TL;DR

This paper explores phase-covariant quantum broadcasting for qubits, showing that relaxing symmetry constraints improves simulation efficiency but remains impractical for observable estimation tasks.

Contribution

It identifies the unique optimal virtual broadcasting map under phase-covariance constraints and compares its efficiency to the unitary covariant case.

Findings

The optimal map minimizes simulation cost within the symmetry family.

The closest physical map is the phase-covariant cloning channel.

Despite improvements, the map remains sample-inefficient and impractical.

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.