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 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.
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.
The efficient and reliable characterization of quantum states plays a vital role in most, if not all, quantum information processing tasks. In this work, we present a universally optimal protocol for verifying entangled states by employing the so-called quantum nondemolition measurements, such that the verification efficiency is equivalent to that of the optimal global strategy. Instead of being probabilistic as the standard verification strategies, our protocol is constructed sequentially, which is thus more favorable for experimental realizations. In addition, the target states are preserved in the protocol after each measurement, so can be reused in any subsequent tasks. We demonstrate the power of our protocol for the optimal verification of Bell states, arbitrary two-qubit pure states, and stabilizer states. We also prove that our protocol is able to perform tasks including fidelity estimation and state preparation.
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 (QSV) is the task of relying on local measurements only to verify that a given quantum device does produce the desired target state. Up to now, certain types of entangled states can be verified efficiently or even optimally by QSV. However, given an arbitrary entangled state, how to design its verification protocol remains an open problem. This study presents a systematic strategy to tackle this problem by considering the locality of what it initiates as the choice‐independent measurement protocols, whose operators can be directly achieved when they are homogeneous. Taking several typical entangled states as examples, this study shows the explicit procedures of the protocol design using standard Pauli projections, demonstrating the superiority of the method for attaining better QSV strategies. Moreover, the framework can be naturally extended to other tasks such as the construction of entanglement witnesses, and even parameter estimation.
Entanglement is an essential resource in many quantum information tasks and the entanglement witness is a widely used tool for its detection. In experiments the prepared state generally deviates from the target state due to some noise. Normally the white-noise model is applied to quantifying such a derivation and at the same time reveals the robustness of the witness. However, there may exist other kinds of noise, in which the coherent noise can dramatically ``rotate'' the prepared state. In this way, coherent noise is likely to lead to a failure of detection, even though the underlying state is actually entangled. In this work, we propose an efficient entanglement detection protocol for N-partite Greenberger-Horne-Zeilinger (GHZ)--like states. The protocol can eliminate the effect of coherent noise and at the same time feedback the corresponding noise parameters, which are beneficial to further improvements of the experiment system. In particular, we consider two experiment-relevant coherent noise models; one is from the unconscious phase accumulation on N qubits, and the other is from the rotation on the control qubit. The protocol effectively realizes a family of entanglement witnesses by postprocessing the measurement results from N+2 local measurement settings, which adds only one more setting than the original witness specialized for the GHZ state. Moreover, by considering the trade-off between the detection efficiency and the white-noise robustness, we further reduce the number of local measurements to three without altering the performance under the coherent noise. Our protocol can enhance entanglement detection under coherent noises and act as a benchmark for state-of-the-art quantum devices.
Entangled two-qubit states are the core building blocks for constructing quantum communication networks. Their accurate verification is crucial to the functioning of the networks, especially for untrusted networks. In this work we study the self-testing of two-qubit entangled states via steering inequalities, with robustness analysis against noise. More precisely, steering inequalities are constructed from the tilted Clauser-Horne-Shimony-Holt inequality and its general form, to verify the general two-qubit entangled states. The study provides a good robustness bound, using both local extraction map and numerical semidefinite-programming methods. In particular, optimal local extraction maps are constructed in the analytical method, which yields the theoretical optimal robustness bound. To further improve the robustness of one-sided self-testing, we propose a family of three measurement settings steering inequalities. The result shows that three-setting steering inequality demonstrates an advantage over two-setting steering inequality on robust self-testing with noise. Moreover, to construct a practical verification protocol, we clarify the sample efficiency of our protocols in the one-sided device-independent scenario.