Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1905.12175 · Physical Review Letters

Experimental Optimal Verification of Entangled States Using Local Measurements

Wen-Hao Zhang, Chao Zhang, Zhe Chen, Xing-Xiang Peng, Xiao‐Ye Xu, Peng Yin, Shang Yu, Xiang-Jun Ye, Yong‐Jian Han, Jin‐Shi Xu, Geng Chen, Chuan‐Feng Li, Guang‐Can Guo

Published Jul 17, 2020• 53 citations
arXivHTMLPDF

Abstract

The initialization of a quantum system into a certain state is a crucial aspect of quantum information science. While a variety of measurement strategies have been developed to characterize how well the system is initialized, for a given one, there is in general a trade-off between its efficiency and the accessible information of the quantum state. Conventional quantum state tomography can characterize unknown states while requiring exponentially expensive time-consuming postprocessing. Alternatively, recent theoretical breakthroughs show that quantum state verification provides a technique to quantify the prepared state with significantly fewer samples, especially for multipartite entangled states. In this Letter, we modify the original proposal to be robust to practical imperfections, and experimentally implement a scalable quantum state verification on two-qubit and four-qubit entangled states with nonadaptive local measurements. For all the tested states, the estimated infidelity is inversely proportional to the number of samples, which illustrates the power to characterize a quantum state with a small number of samples. Compared to the globally optimal strategy which requires nonlocal measurements, the efficiency in our experiment is only worse by a small constant factor (<2.5). We compare the performance difference between quantum state verification and quantum state tomography in an experiment to characterize a four-photon Greenberger-Horne-Zeilinger state, and the results indicate the advantage of quantum state verification in both the achieved efficiency and precision.

DOI
10.1103/physrevlett.125.030506

Related papers

Optimal Verification of Entangled States with Local Measurements

Sam Pallister, Noah Linden, Ashley Montanaro

Consider the task of verifying that a given quantum device, designed to produce a particular entangled state, does indeed produce that state. One natural approach would be to characterize the output state by quantum state tomography, or alternatively, to perform some kind of Bell test, tailored to the state of interest. We show here that neither approach is optimal among local verification strategies for 2-qubit states. We find the optimal strategy in this case and show that quadratically fewer total measurements are needed to verify to within a given fidelity than in published results for quantum state tomography, Bell test, or fidelity estimation protocols. We also give efficient verification protocols for any stabilizer state. Additionally, we show that requiring that the strategy be constructed from local, nonadaptive, and noncollective measurements only incurs a constant-factor penalty over a strategy without these restrictions.

arXiv:1709.03353•Physical Review Letters•Apr 24, 2018•124 citations

Classical communication enhanced quantum state verification

Wen-Hao Zhang, Xiao Liu, Peng Yin +11 more

Abstract Quantum state verification provides an efficient approach to characterize the reliability of quantum devices for generating certain target states. The figure of merit of a specific strategy is the estimated infidelity ϵ of the tested state to the target state, given a certain number of performed measurements n. Entangled measurements constitute the globally optimal strategy and achieve the scaling that ϵ is inversely proportional to n. Recent advances show that it is possible to achieve the same scaling simply with non-adaptive local measurements; however, the performance is still worse than the globally optimal bound up to a constant factor. In this work, by introducing classical communication, we experimentally implement an adaptive quantum state verification. The constant factor is minimized from ~2.5 to 1.5 in this experiment, which means that only 60% measurements are required to achieve a certain value of ϵ compared to optimal non-adaptive local strategy. Our results indicate that classical communication significantly enhances the performance of quantum state verification, and leads to an efficiency that further approaches the globally optimal bound.

arXiv:2010.10011•npj Quantum Information•Dec 15, 2020•18 citations

Verification of Entangled States Under Noisy Measurements

Lan Zhang, Yinfei Li, Ye‐Chao Liu +1 more

Abstract Entanglement plays an indispensable role in numerous quantum information and quantum computation tasks, underscoring the need for efficiently verifying entangled states. In recent years, quantum state verification has received increasing attention, yet the challenge of addressing noise effects in implementing this approach remains unsolved. In this work, a systematic assessment of the performance of quantum state verification protocols is provided in the presence of measurement noise. Based on the analysis, a necessary and sufficient condition is provided to uniquely identify the target state under noisy measurements. Moreover, this work proposes a symmetric hypothesis testing verification algorithm with noisy measurements. Then, relying on states, a semidefinite program is demonstrated to calculate the infidelity threshold for arbitrary measurement noise. Subsequently, using a noisy nonadaptive verification strategy of Greenberger–Horne–Zeilinger and stabilizer states, the noise effects on the verification efficiency are analytically illustrated. From both analytical and numerical perspectives, this work demonstrates that the noisy verification protocol exhibits a negative quadratic relationship between the sample complexity and the infidelity. The method can be easily applied to real experimental settings, thereby demonstrating its promising prospects.

arXiv:2406.01470•Advanced Quantum Technologies•Feb 17, 2025•1 citation

Towards the standardization of quantum state verification using optimal strategies

Xin-he Jiang, Kun Wang, Kaiyi Qian +7 more

Abstract Quantum devices for generating entangled states have been extensively studied and widely used. As so, it becomes necessary to verify that these devices truly work reliably and efficiently as they are specified. Here we experimentally realize the recently proposed two-qubit entangled state verification strategies using both local measurements (nonadaptive) and active feed-forward operations (adaptive) with a photonic platform. About 3283/536 number of copies (N) are required to achieve a 99% confidence to verify the target quantum state for nonadaptive/adaptive strategies. These optimal strategies provide the Heisenberg scaling of the infidelity $${\it{\epsilon }}$$ ϵ as a function of N ( $${\it{\epsilon }}\sim N^{r}$$ ϵ ~ N r ) with the parameter r = −1, exceeding the standard quantum limit with r = −0.5. We experimentally obtain the scaling parameters of r = −0.88 ± 0.03 and −0.78 ± 0.07 for nonadaptive and adaptive strategies, respectively. Our experimental work could serve as a standardized procedure for the verification of quantum states.

arXiv:2002.00640•npj Quantum Information•Oct 27, 2020•32 citations

Quantum memory assisted entangled state verification with local measurements

Siyuan Chen, Wei Xie, Ping Xu +1 more

We consider the quantum memory assisted quantum state verification task, where an adversary prepares independent multipartite entangled states and sends them to the local verifiers, who then store several copies in the quantum memory and measure them collectively to make a decision. We establish an exact analytic formula for optimizing two-copy state verification, where the verifiers store two copies, and give a globally optimal two-copy strategy for multiqubit graph states involving only Bell measurements. When the verifiers can store arbitrarily many copies, we present a dimension expansion technique that designs efficient verification strategies for this case, showcasing its application to efficiently verifying Greenberger-Horne-Zeilinger (GHZ)-like states. These strategies become increasingly advantageous with growing memory resources, ultimately approaching the theoretical limit of efficiency. Our findings demonstrate that quantum memories enhance state verification efficiency, shedding light on error-resistant strategies and practical applications of large-scale quantum memory assisted verification.

arXiv:2312.11066•Physical Review Research•Jan 2, 2025•5 citations

Efficient quantum state tomography with auxiliary systems

Wenlong Zhao, Da Zhang, Huili Zhang +2 more

Quantum state tomography is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields such as quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only makes experimental design and implementation more complex, but also exacerbates the consumption of experimental resources. These limitations severely hinder the application of state tomography in large-scale quantum systems. To reduce measurement settings and improve sampling efficiency, this study proposes a state tomography method based on auxiliary systems. This method can be implemented through either entanglement between the quantum system to be measured and a quantum auxiliary system or through correlation between the quantum system and a probabilistic classical auxiliary system. Measurements on the entire joint system enable more efficient extraction of information about the quantum state to be measured. This method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity of $O(d^2)$, significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for measuring purity based on the proposed circuit, one of which achieves measurement precision at the Heisenberg limit. This study validates the effectiveness of the proposed method through a detailed theoretical analysis, a series of numerical simulations, and experiments.

arXiv:2507.16532•Physical Review A•Mar 2, 2026•0 citations
The initialization of a quantum system into a certain state is a crucial aspect of quantum information science. While a variety of measurement strategies have been developed to characterize how well the system is initialized, for a given one, there is in general a trade-off between its efficiency and the accessible information of the quantum state. Conventional quantum state tomography can characterize unknown states while requiring exponentially expensive time-consuming postprocessing. Alternatively, recent theoretical breakthroughs show that quantum state verification provides a technique to quantify the prepared state with significantly fewer samples, especially for multipartite entangled states. In this Letter, we modify the original proposal to be robust to practical imperfections, and experimentally implement a scalable quantum state verification on two-qubit and four-qubit entangled states with nonadaptive local measurements. For all the tested states, the estimated infidelity is inversely proportional to the number of samples, which illustrates the power to characterize a quantum state with a small number of samples. Compared to the globally optimal strategy which requires nonlocal measurements, the efficiency in our experiment is only worse by a small constant factor (<2.5). We compare the performance difference between quantum state verification and quantum state tomography in an experiment to characterize a four-photon Greenberger-Horne-Zeilinger state, and the results indicate the advantage of quantum state verification in both the achieved efficiency and precision.
Consider the task of verifying that a given quantum device, designed to produce a particular entangled state, does indeed produce that state. One natural approach would be to characterize the output state by quantum state tomography, or alternatively, to perform some kind of Bell test, tailored to the state of interest. We show here that neither approach is optimal among local verification strategies for 2-qubit states. We find the optimal strategy in this case and show that quadratically fewer total measurements are needed to verify to within a given fidelity than in published results for quantum state tomography, Bell test, or fidelity estimation protocols. We also give efficient verification protocols for any stabilizer state. Additionally, we show that requiring that the strategy be constructed from local, nonadaptive, and noncollective measurements only incurs a constant-factor penalty over a strategy without these restrictions.
Abstract Quantum state verification provides an efficient approach to characterize the reliability of quantum devices for generating certain target states. The figure of merit of a specific strategy is the estimated infidelity ϵ of the tested state to the target state, given a certain number of performed measurements n. Entangled measurements constitute the globally optimal strategy and achieve the scaling that ϵ is inversely proportional to n. Recent advances show that it is possible to achieve the same scaling simply with non-adaptive local measurements; however, the performance is still worse than the globally optimal bound up to a constant factor. In this work, by introducing classical communication, we experimentally implement an adaptive quantum state verification. The constant factor is minimized from ~2.5 to 1.5 in this experiment, which means that only 60% measurements are required to achieve a certain value of ϵ compared to optimal non-adaptive local strategy. Our results indicate that classical communication significantly enhances the performance of quantum state verification, and leads to an efficiency that further approaches the globally optimal bound.
Abstract Entanglement plays an indispensable role in numerous quantum information and quantum computation tasks, underscoring the need for efficiently verifying entangled states. In recent years, quantum state verification has received increasing attention, yet the challenge of addressing noise effects in implementing this approach remains unsolved. In this work, a systematic assessment of the performance of quantum state verification protocols is provided in the presence of measurement noise. Based on the analysis, a necessary and sufficient condition is provided to uniquely identify the target state under noisy measurements. Moreover, this work proposes a symmetric hypothesis testing verification algorithm with noisy measurements. Then, relying on states, a semidefinite program is demonstrated to calculate the infidelity threshold for arbitrary measurement noise. Subsequently, using a noisy nonadaptive verification strategy of Greenberger–Horne–Zeilinger and stabilizer states, the noise effects on the verification efficiency are analytically illustrated. From both analytical and numerical perspectives, this work demonstrates that the noisy verification protocol exhibits a negative quadratic relationship between the sample complexity and the infidelity. The method can be easily applied to real experimental settings, thereby demonstrating its promising prospects.
Abstract Quantum devices for generating entangled states have been extensively studied and widely used. As so, it becomes necessary to verify that these devices truly work reliably and efficiently as they are specified. Here we experimentally realize the recently proposed two-qubit entangled state verification strategies using both local measurements (nonadaptive) and active feed-forward operations (adaptive) with a photonic platform. About 3283/536 number of copies (N) are required to achieve a 99% confidence to verify the target quantum state for nonadaptive/adaptive strategies. These optimal strategies provide the Heisenberg scaling of the infidelity ϵ{\it{\epsilon }}ϵ ϵ as a function of N ( ϵ∼Nr{\it{\epsilon }}\sim N^{r}ϵ∼Nr ϵ ~ N r ) with the parameter r = −1, exceeding the standard quantum limit with r = −0.5. We experimentally obtain the scaling parameters of r = −0.88 ± 0.03 and −0.78 ± 0.07 for nonadaptive and adaptive strategies, respectively. Our experimental work could serve as a standardized procedure for the verification of quantum states.
We consider the quantum memory assisted quantum state verification task, where an adversary prepares independent multipartite entangled states and sends them to the local verifiers, who then store several copies in the quantum memory and measure them collectively to make a decision. We establish an exact analytic formula for optimizing two-copy state verification, where the verifiers store two copies, and give a globally optimal two-copy strategy for multiqubit graph states involving only Bell measurements. When the verifiers can store arbitrarily many copies, we present a dimension expansion technique that designs efficient verification strategies for this case, showcasing its application to efficiently verifying Greenberger-Horne-Zeilinger (GHZ)-like states. These strategies become increasingly advantageous with growing memory resources, ultimately approaching the theoretical limit of efficiency. Our findings demonstrate that quantum memories enhance state verification efficiency, shedding light on error-resistant strategies and practical applications of large-scale quantum memory assisted verification.
Quantum state tomography is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields such as quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only makes experimental design and implementation more complex, but also exacerbates the consumption of experimental resources. These limitations severely hinder the application of state tomography in large-scale quantum systems. To reduce measurement settings and improve sampling efficiency, this study proposes a state tomography method based on auxiliary systems. This method can be implemented through either entanglement between the quantum system to be measured and a quantum auxiliary system or through correlation between the quantum system and a probabilistic classical auxiliary system. Measurements on the entire joint system enable more efficient extraction of information about the quantum state to be measured. This method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity of O(d2)O(d^2)O(d2), significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for measuring purity based on the proposed circuit, one of which achieves measurement precision at the Heisenberg limit. This study validates the effectiveness of the proposed method through a detailed theoretical analysis, a series of numerical simulations, and experiments.