It is an intriguing question and certainly something that a number of theorists have been working on. The latest in such a development is the latest paper published in PRA this week, and covered this this week's APS Physics (you also get a free download of the actual paper).
Abstract: We derive quantum theory from purely informational principles. Five
elementary axioms—causality, perfect distinguishability, ideal
compression, local distinguishability, and pure conditioning—define a
broad class of theories of information processing that can be regarded
as standard. One postulate—purification—singles out quantum theory
within this class.
Quantum mechanics continues to be one of the most puzzling and amazing theory of our time.
 G. Chiribella et al., Phys. Rev. A v.84, p.012311 (2011).