Nonlocality without entanglement in exclusion of quantum states

We study the task of quantum state exclusion, focusing on antidistinguishability and its generalization to $x$-antidistinguishability, under global measurements and local operations with classical communication (LOCC). We also introduce weak and strong notions of antidistinguiahbaility ($x$-antidistinguishability) depending on whether all states or all $x$-tuples are exhaustively eliminated. Our results reveal striking differences between state exclusion and the more familiar task of state discrimination. In particular, we show that LOCC antidistinguishability of multipartite product states is symmetric with respect to the initiating party but this symmetry breaks down for higher-order $x$-antidistinguishability. Most notably, we establish a manifestation of \emph{nonlocality without entanglement} in the context of state exclusion: we prove that three bipartite product states can be globally antidistinguishable while failing to be LOCC antidistinguishable, demonstrating that three is the minimal number of states required for this phenomenon. We further extend this separation to $2$-antidistinguishability and present example exhibiting the same type of nonlocality. At last, we provide an antidistinguishable tripartite product states that are not LOCC antidistinguishable across any bipartition, which ensures the phenomenon of \emph{genuine nonlocality without entanglement} in this framework.