No Practical Quantum Broadcasting: Even Virtually

Abstract: Quantum information cannot be broadcast -- an intrinsic limitation imposed by quantum mechanics. However, recent advances in virtual operations have brought new insights into the no-broadcasting theorem. Here, we focus on the practical utility and introduce sample efficiency as a fundamental constraint, requiring any practical broadcasting protocol perform no worse than the naive approach of direct preparation and distribution. We prove that no linear process -- whether quantum or beyond -- can simultaneously uphold sample efficiency, unitary covariance, permutation invariance, and classical consistency. This leads to a no-practical-broadcasting theorem, which places strict limits on the practical distribution of quantum information. To achieve this, we use Schur-Weyl duality to provide a significantly simplified derivation of the uniqueness of the canonical virtual broadcasting map, which satisfies the latter three conditions, and determine its sample complexity via semidefinite programming. Our approach naturally extends the uniqueness of virtual broadcasting to the 1-to-N case and provides its construction. Moreover, we demonstrate that the connection between virtual broadcasting and pseudo-density operators is limited to the 1-to-2 case and generally does not hold, further underscoring the fundamental asymmetry between spatial and temporal statistics in the quantum world.