Explicit Formulas for Estimating Trace of Reduced Density Matrix Powers via Single-Circuit Measurement Probabilities

In the fields of quantum mechanics and quantum information science, the traces of reduced density matrix powers play a crucial role in the study of quantum systems and have numerous important applications. In this study, a universal framework is proposed to simultaneously estimate the traces of the 2nd to the $n$th powers of a reduced density matrix using a single quantum circuit with $n$ copies of the quantum state. Specifically, this approach leverages the controlled SWAP test and establishes explicit formulas connecting measurement probabilities to these traces. Further, two algorithms are developed: a purely quantum method and a hybrid quantum-classical approach combining Newton–Girard iteration. Rigorous analysis via Hoeffding inequality demonstrates the method's efficiency, requiring only $M=O\left(\frac{1}{{\epsilon }^2},\log ,\left(\frac{n}{\delta }\right)\right)$ measurements to achieve precision $\epsilon$ with confidence $1-\delta$. Additionally, various applications are explored including the estimation of nonlinear functions and the representation of entanglement measures. Numerical simulations are conducted for two maximally entangled states, the GHZ state and the W state, to validate the proposed method.