No relative-entropy modification of the Born rule is no-signalling

We consider modifications of the Born rule in which measurement-outcome probabilities are reweighted by a relative-entropy functional of the post-measurement branches. Such entropic selectors are a natural candidate for a dynamical account of outcome selection, and fall outside the scope of steering-based derivations of the Born rule, which cover only convex/affine deformations. We prove a no-go result: under the standard branch-wise prescription, every relative-entropy selector that modifies the Born rule permits superluminal signalling, in every dimension d ≥ 2. The proof reduces, via the Gisin–Hughston–Jozsa–Wootters steering theorem, to the strict non-affinity of the selector, for which a closed-form chord witness is given. We then characterise where the rule is safe: for d = 2 it coincides with the Born rule exactly at the maximally mixed state, whereas for d ≥ 3 it signals at every full-rank state — proven on all diagonal states and an open neighbourhood, with the curvature attaining its global minimum (d−2)/ln d at the maximally mixed state. The conclusion: relative entropy can diagnose and constrain quantum dynamics, but promoting it to an active outcome selector reintroduces signalling. GHJW theorem quantum foundations steering