A lower bound on the classical simulation cost of star-network correlations

It is well established that quantum strategies outperform classical ones in several communication tasks. We study the quantum communication complexity of correlations arising from joint measurements on quantum systems distributed across a star network, where several parties each send a quantum system to a central node. We introduce an exclusion task that can be solved perfectly when each party sends a quantum $d$-level system, but would require a large classical message otherwise. In fact, the task cannot be solved with certainty if each of the $n$ parties sends a classical message with less than $n^{(d-1)}$ symbols. This implies an advantage of using quantum over classical messages in that scenario that scales with both, the dimension of the quantum system and the number of systems measured simultaneously. As an application, this shows that no finite-size classical description of a qubit suffices to reproduce the statistics of a joint measurement on sufficiently many qubits.