Multipartite steering verification with imprecise measurements

TL;DR

This work addresses false positives in multipartite steering verification caused by imprecise measurements. It develops a quantitative framework based on the LHS($T,N$) model and derives a modified inequality that explicitly incorporates imprecision through a bound $B_{\epsilon}$, with a uniform form $B_{\epsilon}=2^{N-T}\left(1+4\sqrt{\epsilon(1-\epsilon)}-8\epsilon\sqrt{\epsilon(1-\epsilon)}\right)^T$. Using GHZ states and depolarized variants, it shows that imprecision reduces the violation weight $W_G$ in an $N$-dependent manner and that relying on ideal bounds can yield false positives; a device-independent comparison confirms the quantitative method provides a more accurate and robust verification range. The results substantially improve the robustness of multipartite steering and entanglement verification under realistic imperfections and offer a generalizable approach for practical quantum technologies including communication and computing.

Abstract

Quantum steering is a fundamental quantum correlation that plays a pivotal role in quantum technologies, but its verification crucially relies on precise measurements -- an assumption often undermined by practical imperfections. Here, we investigate multipartite steering verification under imprecise measurements and develop a quantitative method that effectively eliminates false positives induced by measurement imprecision. A comparison with a device-independent approach demonstrates that our method accurately delineates the scope of valid verification. In a special case, our method also enables the verification of multipartite entanglement under nonideal conditions. These results substantially enhance the robustness of multipartite steering and entanglement verification against measurement imprecision, thereby promoting their applicability in realistic quantum technologies.

Figures (5)

• Figure 1: Schematic illustration of tripartite steering under imprecise measurements. A tripartite state $\rho_{ABC}$ is distributed among Alice, Bob, and Charlie, with respective inputs $\alpha$, $\beta$, and $\gamma$ and corresponding outputs $x$, $y$, and $z$. Alice and Bob are untrusted parties who perform uncharacterized measurements $M_{x|\alpha}$ and $M_{y|\beta}$, while Charlie is the trusted party. In the presence of measurement imprecision $\epsilon$, Charlie’s ideal measurement $M_{z|\gamma}$ is replaced by the imprecise measurement $\tilde{M}_{\tilde{z}|\gamma}^{\epsilon}$, yielding the outcome $\tilde{z}$.
• Figure 2: Simulation of inequality bounds $B$ as functions of the imprecision parameter $\epsilon$ for both ideal and imprecise measurement scenarios. The total number of parties is set to $N=4$, with $T=2$ parties identified as trusted. The solid line represents the modified bound accounting for measurement imprecision, denoted as $B_{\epsilon}$, while the dashed line indicates the original bound for ideal measurements, $B_0$. Shaded areas emphasize the region under each bound, with lighter shading corresponding to $B_{\epsilon}$ and darker shading corresponding to $B_0$.
• Figure 3: Simulation for the violation weights $W_G$ for multipartite steering as a function of the imprecision parameter $\epsilon$, for various values of the total number of parties $N$. Four representative values of $N$ are shown, each distinguished by a different line type. The number of trusted parties is set to $T = \lfloor N/2 \rfloor$. The threshold $W_G=1$ is marked by the horizontal dotted line.
• Figure 4: Simulation for the violation weights $W_G$ for multipartite entanglement as a function of the imprecision parameter $\epsilon$, for different values of the total number of parties $N$. Each value of $N$ is represented by a distinct line style, with the number of trusted parties fixed at $T = N$. The threshold $W_G=1$ is indicated by the horizontal dotted line.
• Figure 5: Simulation of the violation weight $W_d$ as a function of the depolarizing parameter $p$ under different methods. The analysis is performed for the case where $N=4$ and $T=2$. Three typical levels of imprecision parameter are considered: $\epsilon = 0$, $0.5\%$, and $1\%$. Dashed lines represent the violation weight $W_{d,Q}$ obtained via the quantitative method, while dot-dashed lines indicate the violation weight $W_{d,DI}$ for the device-independent method. For each method, a larger value of $\epsilon$ corresponds to a lower line position. The horizontal dotted line marks the threshold $W_d=1$.