The Inverse Born Rule Equivalence. On the Informational Limits of Real-Valued Amplitude Encodings and the Measurement of Quantum Advantage in Data Embeddings

Abstract

When does quantum data encoding provide genuine quantum advantage, and when does it merely rephrase a classically solvable problem? We prove an \emph{Equivalence Theorem} demonstrating that any encoding mapping classical data to real-valued amplitudes, $\vert\psi_c\rangle = \sum_i c_i \vert i\rangle$ with $c_i \in \mathbb{R}$ and $\sum_i c_i^2 = 1$, composed with a data-independent parameterised unitary and computational-basis measurement, yields exactly the class of classical quadratic forms. We identify the geometric mechanism driving this collapse: the restriction to $\mathbb{R}$ forces a vanishing Berry connection, removing the complex phases required for data-dependent quantum interference. To operationalize this boundary, we introduce encoding diagnostics -- phase complexity $C[\Phi]$ and mode-wise von Neumann mutual information $I[\Phi]$ -- and link them to the information-geometric excess $\Delta g$. We show that for all real-valued encodings, $\Delta g = 0$ identically. We term the misidentification of such models as evidence of quantum computational power the \emph{Inverse Born Rule Fallacy}. Supported by numerical experiments, our results establish that complex-phase structure is a strictly necessary condition for data-driven (Type~B) quantum advantage.