Two Results about Quantum Messages

The comparison of communication protocols using quantum messages with protocols using classical messages is a central topic in information and communication theory. It is always good to understand such questions well in the simplest settings where they arise. An example is the setting of one-way communication complexity, which is rich enough to lead to many interesting questions, yet accessible enough for us to prove results about questions like the relationship between different computational modes, e.g. quantum versus classical or nondeterministic versus deterministic.