  Consider a photon that has just emerged from a linear polarizing filter. If
the photon is then subjected to an orthogonal polarization measurement-e.g.,
horizontal vs vertical-the photon's preparation cannot be fully expressed in
the outcome: a binary outcome cannot reveal the value of a continuous variable.
However, a stream of identically prepared photons can do much better. To
quantify this effect, one can compute the mutual information between the angle
of polarization and the observed frequencies of occurrence of "horizontal" and
"vertical." Remarkably, one finds that the quantum-mechanical rule for
computing probabilities--Born's rule--maximizes this mutual information
relative to other conceivable probability rules. However, the maximization is
achieved only because linear polarization can be modeled with a real state
space; the argument fails when one considers the full set of complex states.
This result generalizes to higher dimensional Hilbert spaces: in every case,
one finds that information is transferred optimally from preparation to
measurement in the real-vector-space theory but not in the complex theory.
Attempts to modify the statement of the problem so as to see a similar
optimization in the standard complex theory are not successful (with one
limited exception). So it seems that this optimization should be regarded as a
special feature of real-vector-space quantum theory.
