An efficient method for spot-checking quantum properties with sequential trials

TL;DR

This analysis shows that even as the total number of trials approaches infinity, only a constant number of trials needs to be spot-checked on average to certify the average performance of the remaining trials at a specified confidence level.

Abstract

In practical situations, the reliability of quantum resources can be compromised due to complex generation processes or adversarial manipulations during transmission. Consequently, the trials generated sequentially in an experiment may exhibit non-independent and non-identically distributed (non-i.i.d.) behavior. This non-i.i.d. behavior can introduce security concerns and result in faulty estimates when performing information tasks such as quantum key distribution, self-testing, verifiable quantum computation, and resource allocation in quantum networks. To certify the performance of such tasks, one can make a random decision in each trial, either spot-checking some desired property or utilizing the quantum resource for the given task. However, a general method for certification with a sequence of non-i.i.d. spot-checking trials is still missing. Here, we develop such a method. This method not only works efficiently with a finite number of trials but also yields asymptotically tight certificates of performance. Our analysis shows that even as the total number of trials approaches infinity, only a constant number of trials needs to be spot-checked on average to certify the average performance of the remaining trials at a specified confidence level.

Figures (5)

• Figure 1: Lower confidence bounds $\bar{\Xi}_{{\rm lb}}$ on the average extractability $\bar{\Xi}$ of the Bell state from the past-conditional states generated in the unchecked trials as a function of the true CHSH value $\hat{I}$ underlying the spot-checked trials. We fix the total number of trials at $n=10^5$ and set the confidence level at $(1-\epsilon)=99\;\mathrm{\%}$. Since a separable state can achieve the trivial extractability $0.5$, we truncate all confidence bounds at this value. The upper panel shows the mean of $\bar{\Xi}_{{\rm lb}}$ across $1000$ independent datasets for each method considered. For reference, the asymptotic bound from Ref. kaniewski:2016 is also shown. The inset violin plot shows the distribution of the differences between the confidence bounds obtained by the estimation-factor (EF) method with calibration and the method of Ref. Gocanin2022 for each dataset when $\hat{I}=2.34$ or $2.66$. The lower panel illustrates the mean gap between $\bar{\Xi}_{{\rm lb}}$ and the asymptotic bound. Note that in this figure, as well as in Fig. \ref{['main-fig:finite_efficiency']} below, we also illustrate the performance of the Serfling inequality Serfling1974 solely for comparison. For explanations of the method of Ref. Gocanin2022, the application of the Serfling inequality and how this figure was obtained, see App. \ref{['sect:comparison']}.
• Figure 2: Minimum number of trials $n_{\text{lb}, \min}$ required to ensure that the expected lower confidence bound on $\bar{\Xi}$ exceeds $\check \Xi-\delta_{\text{th}}$, where $\check \Xi=0.9111$ is the asymptotic lower bound on $\bar{\Xi}$ corresponding to a true CHSH value $\hat{I}=2.7$ and $\delta_{\text{th}}$ is the varying expected gap. We set the confidence level to $(1-\epsilon)=99\;\mathrm{\%}$ and assumed i.i.d. trials with spot-checking probability $\omega=0.1$. See App. \ref{['sect:comps']} for details.
• Figure 3: Lower confidence bounds $\bar{\Xi}_{{\rm lb}}$ on the average extractability $\bar{\Xi}$ of the Bell state from the past-conditional states generated in the unchecked trials as a function of the true CHSH value $\hat{I}$. The parameters used for the underlying simulations are the same as those specified for Fig. \ref{['main-fig:tight_bound']} of the main text, except that the spot-checking probability is set to $\omega=0.5$. The confidence bounds were obtained at the confidence level $(1-\epsilon)=99\;\mathrm{\%}$ following the procedure described in this section.
• Figure 4: Minimum number of trials $n_{\text{lb}, \min}$ required to ensure that the expected lower confidence bound on $\bar{\Xi}$ exceeds $\check \Xi-\delta_{\text{th}}$, where $\check \Xi=0.9111$ is the asymptotic lower bound on $\bar{\Xi}$ corresponding to a true CHSH value $\hat{I}=2.7$ and $\delta_{\text{th}}$ is the varying expected gap. We set the confidence level to $(1-\epsilon)=99\;\mathrm{\%}$ and assumed i.i.d. trials with spot-checking probability $\omega=0.5$.
• Figure 5: Lower confidence bounds $\bar{\Xi}_{{\rm lb}}$ on the average extractability $\bar{\Xi}$ of the Bell state from the past-conditional states generated in the unchecked trials as a function of the true CHSH value $\hat{I}$. The parameters used for the underlying simulations are the same as those specified for Fig. \ref{['main-fig:tight_bound']} of the main text, except that we analyze the limit situation $n \to \infty$ and $\omega \to 0$ while the expected number of spot-checked trials is kept constant at $\bar{n}_s=n\omega=10^{4}$. For the plot, we set $n=10^{9}$ after verifying that increasing $n$ further did not significantly affect the results. The confidence bounds were obtained at the confidence level $(1-\epsilon)=99\;\mathrm{\%}$ following the procedure described in this section.

Theorems & Definitions (14)

• Theorem C.1: Estimation-factor theorem
• proof
• Lemma C.2
• proof
• Lemma C.3
• proof
• Lemma E.1
• proof
• Proposition E.2
• proof