Quantifying the Information Gain from Future High-Precision Radius Measurements for Identifying Twin Neutron Stars

Twin neutron stars (NSs), characterized by identical gravitational masses but different radii, are among the most promising astrophysical signatures of a strong first-order hadron--quark phase transition in supradense matter. We investigate how increasingly precise NS radius measurements improve the Bayesian inference of twin-star observability using mock radius data for a canonical $1.4\,M_\odot$ NS. Radius uncertainties are varied from the current level of about $0.9$ km to the $\approx 0.1$ km precision anticipated from future X-ray and gravitational-wave observations. We quantify the information gained using the posterior distribution of the maximum twin-star radius separation $\Delta R$ together with an analytical model of branch distinguishability and complementary information-theoretic measures based on the branch observational efficiency and the Shannon entropy. The combined analyses reveal three inference regimes: a prior-dominated regime for $\sigma_R \gtrsim 0.6$ km, a rapid information-gain regime for $0.2 \lesssim \sigma_R \lesssim 0.6$ km, and an information-saturation regime for $\sigma_R \lesssim 0.2$ km. These complementary analyses consistently indicate that radius measurements with a precision of about $0.2$ km already extract most of the information available for identifying twin NSs within the present Bayesian framework. Beyond establishing a quantitative observational benchmark for future high-precision radius measurements, this work provides a general Bayesian framework for quantifying the information gain from progressively more precise observations and identifying the point of diminishing scientific returns.