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Imprecise quantum steering inequalities in tripartite systems

Yan Zhao, Li-Juan Li, Zheng-Peng Xu, Liu Ye, Dong Wang

Abstract

Quantum steering, as a manifestation of nonlocal quantum correlations, plays a crucial role in enabling various quantum information processing tasks. However, practical implementations are often hindered by significant challenges arising from imperfect or untrusted measurement devices. This study investigates the impact of measurement inaccuracies on quantum steering, with a particular focus on errors in the untrusted party's measurement devices. We first analyze how such errors affect the evaluation of steering inequalities, and then derive bipartite steering inequalities based on correlation matrices under imperfect measurements. Our findings show that even small measurement errors can significantly compromise the certification of quantum steerability, an effect that becomes particularly pronounced as the system dimension increases. Furthermore, by extending the proposed steering inequality to a modified tripartite scenario via correlation matrices, we demonstrate that the influence of measurement imperfections is far more severe in multipartite quantum steering than in the bipartite case. Our results underscore the critical need to account for measurement imperfections in experimental quantum steering and provide a theoretical framework for characterizing and mitigating these effects in high-dimensional quantum systems.

Imprecise quantum steering inequalities in tripartite systems

Abstract

Quantum steering, as a manifestation of nonlocal quantum correlations, plays a crucial role in enabling various quantum information processing tasks. However, practical implementations are often hindered by significant challenges arising from imperfect or untrusted measurement devices. This study investigates the impact of measurement inaccuracies on quantum steering, with a particular focus on errors in the untrusted party's measurement devices. We first analyze how such errors affect the evaluation of steering inequalities, and then derive bipartite steering inequalities based on correlation matrices under imperfect measurements. Our findings show that even small measurement errors can significantly compromise the certification of quantum steerability, an effect that becomes particularly pronounced as the system dimension increases. Furthermore, by extending the proposed steering inequality to a modified tripartite scenario via correlation matrices, we demonstrate that the influence of measurement imperfections is far more severe in multipartite quantum steering than in the bipartite case. Our results underscore the critical need to account for measurement imperfections in experimental quantum steering and provide a theoretical framework for characterizing and mitigating these effects in high-dimensional quantum systems.
Paper Structure (6 sections, 3 theorems, 49 equations, 3 figures)

This paper contains 6 sections, 3 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Einstein-Podolsky-Rosen steering
  3. Imprecise measurements
  4. steering inequality with Imprecise measurements
  5. Discussion and Conclusion
  6. Acknowledgments

Key Result

Proposition 1

Consider a bipartite state $\hat{\rho}_{AB}$ where Alice performs uncharacterized measurements ${\hat{A}_i}$, while Bob attempts to perform a trusted set of projective measurements ${\hat{B}_i}$ but instead implements imperfect observables ${\tilde{\hat{B}}_i}$. Assume that the deviation between eac where $\tilde{\Lambda}_{b}\leq\Lambda_{b}+2\sum_{i=1}^{m}\eta[1+\mathrm{Tr}(\hat{B}_{i}\hat{\rho}_{

Figures (3)

  • Figure 1: The dependence of the steerability parameter $p$ of a quantum state on the magnitude of experimental error, quantified in the range $\xi \in [0,10^{-4}]$. Here, $p$ represents the critical threshold for certifying quantum steering, with values $p \geq 0.577$ ($A\rightarrow{B}$) and $p \geq 0.565$ ($B\rightarrow{A}$) indicating steerable states under ideal conditions. As the error $\xi$ increases from 0 to 0.01$\%$, the parameter $p$ monotonically increases to 1, demonstrating a degradation of steerability with rising imperfections.
  • Figure 2: The quantum steerability $H_{A \rightarrow BC}$ with respect to measurement imperfection $\xi$ and the state's parameter $\theta$. The black solid line represents the zero-valued steering (i.e., $H_{A \rightarrow BC}=0$). The black solid line denotes the critical steering boundary. To be explicit, the states are steerable in the regions of below the boundary, contrarily, the systems are unsteerable in the regions of above the boundary. When $\xi$ exceeds a critical value, the system becomes completely unsteerable which is independent of $\theta$.
  • Figure 3: Comparison of dimension-dependent error sensitivity in quantum steering protocols. (a) Bipartite systems ($A \rightarrow B$): Steering detectability degrades with increasing measurement error $\xi$, with higher-dimensional systems showing accelerated degradation. (b) Tripartite $A \rightarrow BC$ steering: the joint subsystem $BC$ (effectively $d^2$-dimensional) introduces amplified noise propagation, making this configuration particularly vulnerable to measurement imperfections. (c) Tripartite $BC \rightarrow A$ steering: the error effects are confined to the single untrusted subsystem $A$, leading to robustness against quantum imperfection in comparison with the case in Graph (b).

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3