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

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.

arXiv comment29 pages, 19 figures

References in corpus (29)

• Quantum entanglement
• Quantum Cryptography
• Entanglement of Formation of an Arbitrary State of Two Qubits
• Quantum Communication Through an Unmodulated Spin Chain
• Quantum principal component analysis
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Measurement-based quantum computation
• Direct estimations of linear and non-linear functionals of a quantum state
• Universal Quantum Estimator
• Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality
• Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation
• Estimating entanglement measures in experiments
• Entanglement of Formation of Bipartite Quantum States
• Multipartite entanglement measures
• Dynamics of quantum systems
• General Monogamy of Tsallis $q$-Entropy Entanglement in Multiqubit Systems
• Von Neumann entropy from unitarity
• Entanglement spectroscopy on a quantum computer
• Quantify entanglement by concurrence hierarchy
• Entanglement spectroscopy with a depth-two quantum circuit
• Detecting entanglement in quantum many-body systems via permutation moments
• Qubit-efficient entanglement spectroscopy using qubit resets
• A new parameterized entanglement monotone
• Variational Gibbs State Preparation on NISQ devices
• Parameterized multipartite entanglement measures
• Multipartite entanglement measures via Bell basis measurements
• Calculating Concentratable Entanglement in Graph States
• A Hierarchy of Multipartite Correlations Based on Concentratable Entanglement
• Parameterized bipartite entanglement measures and entanglement constraints