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

TL;DR

This paper introduces a universal quantum measurement framework that efficiently estimates traces of reduced density matrix powers using a single circuit, with applications in quantum entanglement and nonlinear function estimation.

Contribution

It provides explicit formulas and algorithms for estimating multiple density matrix traces simultaneously with a single measurement circuit, improving efficiency over previous methods.

Findings

Requires only O(1/ε^2 log(n/δ)) measurements for desired precision

Validates approach with numerical simulations on GHZ and W states

Establishes explicit formulas linking measurement probabilities to density matrix traces

Abstract

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 paper, we propose a universal framework to simultaneously estimate the traces of the $2$nd to the $n$th powers of a reduced density matrix using a single quantum circuit with $n$ copies of the quantum state. Specifically, our approach leverages the controlled SWAP test and establishes explicit formulas connecting measurement probabilities to these traces. We further develop two algorithms: 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 $M=O\left(\frac{1}{\epsilon^2}\log(\frac{n}{\delta})\right)$ measurements to achieve precision $\epsilon$ with confidence $1-\delta$. Additionally, we explore various applications 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.