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arXiv:2002.00640 · npj Quantum Information

Towards the standardization of quantum state verification using optimal strategies

Xin-he Jiang, Kun Wang, Kaiyi Qian, Zhaozhong Chen, Zhiyu Chen, Liangliang Lu, Lijun Xia, Fangmin Song, Shining N. Zhu, Xiao‐Song Ma

Published Oct 27, 2020• 32 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41534-020-00317-7

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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.
Abstract High-dimensional entanglement is significant for the fundamental studies of quantum physics and offers unique advantages in various quantum information processing tasks. Integrated quantum devices have recently emerged as a promising platform for creating, processing, and detecting complex high-dimensional entangled states. A crucial step toward practical quantum technologies is to verify that these devices work reliably with an optimal strategy. In this work, we experimentally implement an optimal quantum verification strategy on a three-dimensional maximally entangled state using local projective measurements on a silicon photonic chip. A 95% confidence is achieved from 1190 copies to verify the target quantum state. The obtained scaling of infidelity as a function of the number of copies is −0.5497 ± 0.0002, exceeding the standard quantum limit of −0.5 with 248 standard deviations. Our results indicate that quantum state verification could serve as an efficient tool for complex quantum measurement tasks.
Abstract The efficient and reliable verification of quantum states plays a crucial role in various quantum information processing tasks. We consider the task of verifying entangled states using one-way and two-way classical communication and completely characterize the optimal strategies via convex optimization. We solve these optimization problems using both analytical and numerical methods, and the optimal strategies can be constructed for any bipartite pure state. Compared with the nonadaptive approach, our adaptive strategies significantly improve the efficiency of quantum state verification. Moreover, these strategies are experimentally feasible, as only few local projective measurements are required.
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.
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.