The Born Rule for Projective Measurements from Metric Non-Expansion and Calibration

Abstract

Why should a calibrated quantum measurement report Born probabilities? Fix a finite dimension $d\geq 2$ and a projective measurement $M$ on $\mathbb{C}^d$, and let $P_M$ be a map assigning outcome probabilities to pure states. I prove that three per-apparatus hypotheses force $P_M$ to be the Born readout: the square-root readout $R_M=\sqrt{P_M}$ is absolutely continuous along Fubini--Study geodesics; the classical Fisher speed of the output never exceeds the quantum Fisher speed of the input almost everywhere on smooth pure-state curves; and the readout reports certainty on every state in each labeled eigenspace. Taking square roots turns probabilities into coordinates on a spherical orthant, and calibration pins the vertices. The metric hypotheses make $R_M$ globally $1$-Lipschitz from Fubini--Study to round distance, so the readout cannot move farther from any calibrated vertex: every coordinate is at least its Born value, both vectors have unit norm, and they are equal. The hypotheses are per measurement; imposing them in every projective context yields the Born assignment on all projective measurements, at every $d\geq 2$, with noncontextuality arriving as an output rather than an axiom. The metric argument stops at projective measurements: for every non-projective POVM, an explicit family of non-Born readouts satisfies the same metric hypotheses even with the stated certainty-of-occurrence calibration.