Power and Limits of Collective Local Measurements in Multicopy State Discrimination

Abstract

More than two decades ago, Bennett \emph{et al.} [Phys. Rev. A \textbf{59}, 1070 (1999)] asked whether perfect local discrimination of orthogonal quantum states can require more than two copies. This question was subsequently answered for adaptive protocols that process the copies separately [Phys. Rev. Lett. \textbf{126}, 210505 (2021)], but remained open when each laboratory is allowed to process its local copies collectively. Here we resolve this stronger setting and identify collective access across repeated local inputs as a distinct resource. For every fixed odd-prime local dimension, there exist complete maximal-stabilizer eigenbases whose copy complexity under individual-copy separable measurements diverges with system size. Under collective processing this behavior changes sharply: every maximal-stabilizer eigenbasis in odd-prime local dimension is perfectly decoded by one-round collective LOCC using at most three copies. Collective processing, however, does not remove multicopy hardness in general. For every fixed local dimension $d\ge2$, we prove the existence, within an explicit phase family, of complete bases whose copy complexity remains unbounded even under collective separable measurements, with a square root of the number of subsystems as lower-bound scale. Thus sample number, spatial measurement power, and coherent access across repeated local inputs are distinct resources in distributed quantum measurement.