Optimal Information Transfer and Real-Vector-Space Quantum Theory

Understand

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.