Bounding the classical cost of simulating quantum behaviors in the prepare-and-measure scenario

We study the prepare-and-measure scenario in which Alice transmits a quantum system to Bob, who then performs a quantum measurement. The quantum state of the system is unknown to Bob, and the measurement is unknown to Alice. It has recently been shown that shared randomness and two bits of classical communication are necessary and sufficient to simulate the transmission of a qubit. We show that the communication cost can be reduced to an average of $1.89$ bits. We then study restricted sets of state preparations: First, for a restriction to real-valued qubit states, if the communication of a classical trit is sufficient, we show that the corresponding protocol must have a convoluted form. We then reduce the smallest qubit scenario requiring two bits of classical communication to only $6$ state preparations and $5$ measurements. For a qutrit, it is not known whether the communication cost is finite; we identify a scenario that requires at least $5$ classical messages, already for the simulation of the real qutrit. Finally, we develop a method for restricted sets of states, that allows us to lower bound the classical communication cost based solely on the set of quantum states.