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npj Quantum Information

Few-shot estimation of entanglement with Bell measurement assistance

Gong-Chu Li, Lei Chen, Xu-Song Hong, Si-Qi Zhang, Huaqing Xu, Yuancheng Liu, You Zhou, Geng Chen, Chuan-Feng Li, Guang‐Can Guo

Published Jan 9, 2026• 1 citation
Publisher

Abstract

Entanglement is fundamental to quantum physics and information processing. In this work, we introduce the Few-Shot Randomized Measurement (FSRM) method, developing an unbiased estimator for mixed-state entanglement from just three experimental shot outcomes. By incorporating the Bell measurement (BM), we supplement the traditional computational-basis measurement to enhance the randomized measurement scheme, which is scalable to n-qubit systems via BMs on qubit pairs. Our approach enables direct estimation of entanglement through random unitary evolution in a photonic system. Compared to the classical shadow method, BM-enhanced FSRM requires no prior knowledge of the local unitaries, offering greater robustness against unitary imperfections. Additionally, we find that utilizing more versatile measurement settings with fewer repeats per setting is more efficient under fixed measurement resources. Our protocol and experimental demonstration represent a significant advancement in the efficient and practical characterization of quantum states.

DOI
10.1038/s41534-025-01172-0

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Entanglement is fundamental to quantum physics and information processing. In this work, we introduce the Few-Shot Randomized Measurement (FSRM) method, developing an unbiased estimator for mixed-state entanglement from just three experimental shot outcomes. By incorporating the Bell measurement (BM), we supplement the traditional computational-basis measurement to enhance the randomized measurement scheme, which is scalable to n-qubit systems via BMs on qubit pairs. Our approach enables direct estimation of entanglement through random unitary evolution in a photonic system. Compared to the classical shadow method, BM-enhanced FSRM requires no prior knowledge of the local unitaries, offering greater robustness against unitary imperfections. Additionally, we find that utilizing more versatile measurement settings with fewer repeats per setting is more efficient under fixed measurement resources. Our protocol and experimental demonstration represent a significant advancement in the efficient and practical characterization of quantum states.
Entanglement plays a fundamental role in quantum physics and information processing. Here, we develop an unbiased estimator for mixed-state entanglement in the few-shot scenario and directly estimate it using random unitary evolution in a photonic system. As a supplement to traditional projective measurements, we incorporate Bell measurements on qubit-pairs, enriching the previous randomized measurement scheme, which is no-go in this task with only local unitary evolution. The scheme is scalable to n-qubits via Bell measurements on qubit-pairs. The estimator can be derived directly from a few consecutive outcomes while exhibiting greater robustness to system errors and noise compared to schemes based on shadow estimation. We find that, under a fixed measurement resource, the way with more versatile measurement settings with fewer repeats per setting is more efficient. Our protocol and demonstration advance the direct characterization of quantum states in practice.
We present a technique for enhancing the estimation of quantum state properties by incorporating approximate prior knowledge about the quantum state. This consists in performing randomized measurements on a quantum processor and comparing the results with those obtained from a classical computer that stores an approximation of the quantum state. We provide unbiased estimators for expectation values of multicopy observables and present performance guarantees in terms of variance bounds that depend on the prior knowledge accuracy. We demonstrate the effectiveness of our approach through experimental and numerical examples detecting mixed-state entanglement, and estimating polynomial approximations of the von Neumann entropy and state fidelities. Published by the American Physical Society 2024
There have been recently significant progress in quantum technologies to realize physically diverse quantum computation and simulation platforms.The aim of this thesis is to develop theoretical approaches based on measurement protocols to experimentally characterize entanglement generated in these state-of-art quantum technological platforms. In the past few years, protocols belonging to the "randomized measurements toolbox" (RM toolbox) have shown to have the ability to measure bipartite entanglement (entanglement between two parts of a quantum system). These protocols have been implemented in several experimental platforms consisting of quantum simulators and quantum computers beyond the regime of full state tomography.The aim of this manuscript is principally dedicated to the development of new approaches based on the RM toolbox to provide experimental groups with advanced tools and methods to probe new fundamental quantities of interest related to entanglement prepared in large quantum platforms.The first objective of this thesis is to provide a new protocol to optimize the existing RM protocol by reducing the statistical errors in the estimation of the quantities. This is done with two approaches: (i) via importance sampling of local random unitaries and (ii) by implementing common randomized measurements. These techniques automatically enable us to implement the RM toolbox to estimate properties associated to entanglement on larger experimental platforms. Then, a central part of this work proposes an alternate effective protocol to measure a fundamental quantity in quantum metrology known as the "quantum Fisher information" that enables us to characterize the multipartite entanglement content of the underlying quantum state. This novel protocol is based on estimating the QFI via the RM toolbox, as a converging series of lower bound polynomials of the density matrix.Additionally, the manuscript also focuses on solving a practical bottleneck of the current RM toolbox caused by the over-burdening classical cost to post-process experimental RM data to evaluate unbiased estimations of relevant quantities.This motivates the introduction of the "batch shadow formalism" that circumvents this problem and provides an efficient data treatment technique with rigorous performance guarantees to get estimates of quantities accessible from the experimental RM data. Lastly, owing to these developments of the RM toolbox, the manuscript reports estimation of some quantities that were accessed for the first time by experimental implementation of this toolbox on two different architectures of quantum platform consisting of superconducting qubits and trapped ions.
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Entanglement plays a central role in quantum information processing and quantum physics. However, few effective ways are known to detect the amount of entanglement of an unknown quantum state. Here, we propose a scheme to estimate the entanglement negativity for any bipartition of a composite system. The proposed scheme is based on the random unitary evolution and local measurements on a single-copy quantum state, which is more practical compared to former methods based on collective measurements on many copies of the identical state. Meanwhile, we generalize the scheme to quantify the total correlation. We demonstrate the efficiency of the scheme with statistical analyses and numerical simulations. Our scheme is quite suitable for state-of-the-art quantum platforms, which can serve as a useful benchmarking tool to advance quantum technologies and a probe to study fundamental quantum physics like entanglement dynamics.