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Charting the Shape of Quantum State Space


Physicists have become accustomed to the idea that a theory's content is always most transparent when written in coordinate-free language.  But sometimes the choice of a good coordinate system can be quite useful for settling deep conceptual issues.  This is particularly so for an information-oriented or Bayesian approach to quantum foundations:  One good coordinate system may (eventually!) be worth more than a hundred blue-in-the-face arguments.  This talk will motivate and chronicle the search for one such class of coordinate systems for finite dimensional operator spaces, the so-called Symmetric Informationally Complete measurements.  The desired class will take little more than five minutes to define, but the quest to construct these objects will carry us down a 35 year journey, with the most frenzied activity only recently.  If time permits, I will turn the tables and discuss how one might hope to get the formal content of quantum mechanics out of the very existence of such a coordinate system.