Timelike correlations and quantum tensor product structure

The state-space structure for a composite quantum system is postulated among several mathematically consistent possibilities that are compatible with a local quantum description. For instance, the unentangled Gleason's theorem allows a state space that includes density operators as a proper subset among all possible composite states. However, bipartite correlations obtained in Bell-type experiments from this broader state space are, in fact, quantum simulable [Barnum et al., Phys. Rev. Lett. 104, 140401 (2010)], and hence, such spacelike correlations are no good for making a distinction among different compositions. In this work we analyze the communication utilities of these different composite models and show that they can lead to distinct utilities in a simple communication game involving two players. Our analysis thus establishes that a beyond quantum composite structure can lead to beyond quantum correlations in the timelike scenario and hence welcomes new principles to isolate the quantum correlations from the beyond quantum ones. We also prove a no-go theorem that the classical information carrying capacity of different such compositions cannot be greater than that of the corresponding quantum composite systems.