Specifying the Intrinsic Back-action of a General Measurement

Abstract

Understanding the invasive nature of quantum measurement and its implications in quantum foundations and information science demands a mathematically rigorous and physically well-grounded characterization of intrinsic back-action in general measurement processes. However, such a framework remains elusive, leaving a critical gap in quantum theory. Here, we address this issue by conceptualizing a general quantum measurement as a reduction of extended projection measurements ensured by Naimark's theorem and, derive a state-updating rule for the concerned measurement as a reduction of the projective measurements postulate. Our framework provides a detailed analysis by explicitly decomposing the disturbance effects into two distinct contributions: those arising from the measurement elements themselves and those resulting from the dilation process. Notably, this formulation naturally recovers the projection postulate in the case of projective measurements. Beyond providing insights into joint measurability, non-disturbance, our rule establishes quantitaive connections between intrinsic disturbance and other fundamental quantum features, such as randomness, uncertainty, and information gain.