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Weak measurements of photon position can be used to obtain directexperimental evidence of the wavefunction of a photon between generation andultimate detection. Significantly, these measurement results can also beunderstood as complex valued conditional probabilities of intermediate photonpositions. It is therefore possible to interpret the quantum state as a complexvalued probability distribution from which measurement probabilities can bederived according to Bayesian rules. The conventional measurement probabilitiesderived from squares of the wavefunction then describes the effects ofmeasurement back-action, which originate from a non-classical relation betweendynamics and statistics that is characteristic of quantum mechanics. It ispointed out that this relation can be used to derive the complete Hilbert spaceformalism directly from complex probabilities, without the axiomaticintroduction of quantum states or operators.

Quantum states as complex probabilities: The physics behind direct observations of photon wavefunctions in weak measurements