Unveiling Entanglement's Metrological Power: Empirical Modeling of Optimal States in Quantum Metrics

Abstract

Using extensive numerical analysis of 20,000 randomly generated two-qubit states, we provide a quantitative analysis of the connection between entanglement measures and Maximized Quantum Fisher Information (MQFI). Our systematic study shows strong empirical relationships between the metrological capacity of quantum states and three different entanglement measures: concurrence, negativity, and relative entropy of entanglement. We show that optimization over local unitary transformations produces substantially more predictable relationships than fixed-generator quantum Fisher information approaches using sophisticated statistical analysis, such as bootstrap resampling, systematic data binning, and multiple model comparisons. With exponential fits reaching $R^2 > 0.99$ and polynomial models reaching $R^2 = 0.999$, we offer thorough empirical support for saturation behavior in quantum metrological advantage. With immediate applications to realworld quantum sensing protocols, our findings directly empirically validate important predictions from quantum resource theory and set fundamental bounds for quantum sensor optimization and resource allocation. These intricate relationships are quantitatively described by the polynomial and exponential fit equations, which offer crucial real-world direction for the design of quantum sensors.