Operational Derivation of the Born Rule and Projective Update in the Enter Calculus

Résumé fourni par la source

The Enter calculus is a linear quantum programming language in which measurement is represented by an explicit operation that consumes quantum resources and returns classical records. This paper studies abstract evaluation strategies for that operation. Five operational requirements are imposed: composability of measurements on disjoint subsystems, no-signaling, nullity of outcomes orthogonal to the state, continuity, and equivalence between immediate evaluation and coherent record creation followed by delayed readout. The main result is that these requirements determine both Born probabilities and projective state updates. The central step is a derivation of diagonal anchoring rather than an assumption of it. No-signaling first implies that a local outcome law depends only on the reduced density operator. Equal-amplitude states with distinct correlated labels are then shown to have uniform outcome laws. A naïve fine-graining argument fails because its refined register retains intra-block coherence. The repaired construction exports a within-block index to an additional register, producing distinct companion states for every refined label. Uniformity and fiber counting then determine all rational diagonal probabilities, while continuity extends the result to the general case. The record principle transfers this result to arbitrary coherent states, and comparison with all later experiments fixes the residual state. The results extend to mixed states, adaptive protocols, rotated projective measurements, POVM outcome laws, and, after adding explicit discard, quantum instruments. A restricted compatibility theorem for sharp contexts is applied to the Frauchiger–Renner protocol. The paper also distinguishes this operational result from the static functional equation studied in a companion work. Keywords: Born rule, quantum measurement, quantum programming languages, linear types, no-signaling, deferred measurement, quantum instruments, Frauchiger–Renner paradox.

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