Device-independent point estimation from finite data and its application to device-independent property estimation

The device-independent approach to physics is one where conclusions are drawn directly from the observed correlations between measurement outcomes. In quantum information, this approach allows one to make strong statements about the properties of the underlying systems or devices solely via the observation of Bell-inequality-violating correlations. However, since one can only perform a finite number of experimental trials, statistical fluctuations necessarily accompany any estimation of these correlations. Consequently, an important gap remains between the many theoretical tools developed for the asymptotic scenario and the experimentally obtained raw data. In particular, a physical and concurrently practical way to estimate the underlying quantum distribution has so far remained elusive. Here, we show that the natural analogs of the maximum-likelihood estimation technique and the least-square-error estimation technique in the device-independent context result in point estimates of the true distribution that are physical, unique, computationally tractable, and consistent. They thus serve as sound algorithmic tools allowing one to bridge the aforementioned gap. As an application, we demonstrate how such estimates of the underlying quantum distribution can be used to provide, in certain cases, trustworthy estimates of the amount of entanglement present in the measured system. In stark contrast to existing approaches to device-independent parameter estimations, our estimation does not require the prior knowledge of any Bell inequality tailored for the specific property and the specific distribution of interest.

Figure 1

Device-independent (DI) parameter estimation through the regularization of experimentally determined relative frequency $\vec{f}$. Here, ${\tilde{\mathcal{Q}}}_{\ell }$ denotes an outer approximation of the quantum set $\mathcal{Q}$. The dashed lines show an alternative path to device-independent estimation using the strength of the observed Bell violation. Here, the quality of estimates relies crucially on the choice of Bell inequalities (see Fig. 2 for an example). In contrast, we obtain, as a byproduct in our approach, an optimized device-independent witness (a Bell-like inequality) for the corresponding parameter estimation. See Appendix pp7 for the connection between such a witness and an ordinary Bell inequality as well as the discussion at Sec. 6 for subtleties involved in using the Bell violation of $\vec{f}$ for device-independent parameter estimations.

Figure 2

Mean value of the normalized negativity estimated from the regularized distributions (average over ${10}^4$ runs) as a function of $N_{\text{trials}}$ based on the relative frequencies $\vec{f}$ generated from ${\vec{P}}_{\mathcal{Q}}^{{\tau }_{1.25}}$ (see Appendix pp5-s1), which has $\mathrm{N}\left({\vec{P}}_{\mathcal{Q}}^{{\tau }_{1.25}}\right)\approx 0.38$. In estimating the negativity, we feed the distributions regularized to ${\tilde{\mathcal{Q}}}_2$ into the second level SDP of [19]. The lower and upper limit of each “error bar” mark, respectively, the 10% and 90% window of the spread of the negativity value. Inset: The corresponding histograms for $N_{\text{trials}}={10}^6$. On average, the ML estimator performs considerably better than the LS estimator.

Figure 3

Plots of the mean value of the 1-norm deviation $\left|\right|{\vec{P}}_{\text{Reg}}\left(\vec{f}\right)-{\vec{P}}_{\mathcal{Q}}{\left|\right|}_1$, average over ${10}^4$ runs, based on various ${\vec{P}}_{\mathcal{Q}}$ as a function of the number of trials $N_{\text{trials}}$ for the projection method (Projection), the modified least-square (LS) method with the nonsignaling polytope as the target set (${\mathrm{LS}}_{\mathcal{N}}$), the least-square method (LS), the modified ML method with a minimization to the nonsignaling polytope (${\mathrm{ML}}_{\mathcal{N}}$) and the ML method (ML). For details on the ${\mathrm{LS}}_{\mathcal{N}}$ and the ${\mathrm{ML}}_{\mathcal{N}}$ method, see Appendix pp8. From top (left to right) to bottom (left to right), we have the plots based on ${\vec{P}}_{\mathcal{Q}}^{\text{CHSH}}$, the plots based on ${\vec{P}}_{\mathcal{Q}}^{\text{90\%CHSH}}$, the plots based on ${\vec{P}}_{\mathcal{Q}}^{{\tau }_{1.25}}$, and the plots based on ${\vec{P}}_{\mathcal{Q}}^{\text{MDL}}$ (see Appendix pp5-s1 for details about these quantum distributions). In each subfigure, the corresponding inset shows a zoom-in view of the plots for $N_{\text{trials}}={10}^6$.

Figure 4

Plots of the mean value of the normalized Bell violations computed from ${\vec{P}}_{\text{Reg}}\left(\vec{f}\right)$ over ${10}^4$ relative frequencies $\vec{f}$ generated from various ${\vec{P}}_{\mathcal{Q}}$ as a function of the number of trials $N_{\text{trials}}$. From top (left to right) to bottom (left to right), we have the plots based on ${\vec{P}}_{\mathcal{Q}}^{\text{CHSH}}$ and ${\vec{P}}_{\mathcal{Q}}^{\text{90\%CHSH}}$ in conjunction with the CHSH Bell inequality [Eq. (E1)], the plots based on ${\vec{P}}_{\mathcal{Q}}^{{\tau }_{1.25}}$ and the Bell inequality ${\mathcal{I}}_{\tau }$ with $\tau =1.25$ [Eq. (E4)], as well as the plots based on ${\vec{P}}_{\mathcal{Q}}^{\text{MDL}}$ and the MDL inequality [Eq. (E6)] with $P(x,y)=\frac{1}{4}$ for all $x,y$ and $l=0.1$, $h=1-3l=0.3$. For details about the quantum distributions considered, see Appendix pp5-s1. The lower and upper limit of each error bar mark, respectively, the 10% and 90% window of the spread of the values plotted. In each subfigure, the inset shows the corresponding log-log plot of the absolute value of the mean value.

Figure 5

Plots of the mean squared error of the Bell violations corresponding to those described in Fig. 4 over ${10}^4$ relative frequencies $\vec{f}$ generated from various ${\vec{P}}_{\mathcal{Q}}$ as a function of the number of trials $N_{\text{trials}}$. From top (left to right) to bottom (left to right), we have the plots based on ${\vec{P}}_{\mathcal{Q}}^{\text{CHSH}}$ and ${\vec{P}}_{\mathcal{Q}}^{\text{90\%CHSH}}$ in conjunction with the CHSH Bell inequality [Eq. (E1)], the plots based on ${\vec{P}}_{\mathcal{Q}}^{{\tau }_{1.25}}$ and the Bell inequality ${\mathcal{I}}_{\tau }$ with $\tau =1.25$ [Eq. (E4)], as well as the plots based on ${\vec{P}}_{\mathcal{Q}}^{\text{MDL}}$ and the MDL inequality [Eq. (E6)] with $P(x,y)=\frac{1}{4}$ for all $x,y$ and $l=0.1$, $h=1-3l=0.3$. For details about the quantum distributions considered, see Appendix pp5-s1. The lower and upper limit of each error bar mark, respectively, the 10% and 90% window of the spread of the values plotted. In each subfigure, the inset shows histograms of the normalized Bell violations for $N_{\text{trials}}={10}^6$.

Figure 6

A two-dimensional slice in the space of distributions. The point ${\vec{P}}_{\mathcal{Q}}^{{\tau }_{1.25}}$ is the distribution considered in the study of negativity. The deterministic point ${\vec{P}}_{11}$ corresponds to $P(1,1|x,y)=1$ for all $x,y$. The product distribution $P_{\text{prod}}(a,b|x,y)=P_{\mathcal{Q}}^{{\tau }_{1.25}}\left(a\right|x)P_{\mathcal{Q}}^{{\tau }_{1.25}}\left(b\right|y)$ is obtained from the marginals of $P_{\mathcal{Q}}^{{\tau }_{1.25}}$. In black and red, respectively, we have the boundary of the local and the nonsignaling polytopes $\mathcal{L}$ and $\mathcal{N}$. We also plot, for outer approximations of level $\ell =1$ (blue) and $\ell =2$ (magenta), the approximated quantum set ${\tilde{\mathcal{Q}}}_{\ell }$, and sets of upper-bounded negativity ${\tilde{\mathcal{Q}}}_{\ell }^{\le \nu }$ for $\nu =0.2$ and 0.3.