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arXiv:2010.09442 · Science China Information Sciences

Experimental entanglement quantification for unknown quantum states in a semi-device-independent manner

Yu Chen Guo, Lijinzhi Lin, Huan Cao, Chao Zhang, Xiaodie Lin, Xiao‐Min Hu, Bi‐Heng Liu, Yun‐Feng Huang, Zhaohui Wei, Yong‐Jian Han, Chuan‐Feng Li, Guang‐Can Guo

Published Jul 4, 2023• 4 citations
arXivHTMLPDF
DOI
10.1007/s11432-022-3681-2

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We define a property called nondegeneracy for Bell inequalities, which describes the situation that in a Bell setting, if a Bell inequality and involved local measurements are fixed, any quantum state with a given dimension and its orthogonal quantum state cannot violate remarkably the inequality simultaneously. By choosing a proper nondegenerate Bell inequality, we prove that for an unknown bipartite quantum state of given dimension, based on the measurement statistics only, we can provide an analytic lower bound for the entanglement of formation or even the distillable entanglement, making the whole process semi device independent. We characterize the mathematical structure of nondegeneracy, and prove that quite a lot of well-known Bell inequalities are nondegenerate. We demonstrate our approach by quantifying entanglement for qutrit-qutrit states based on their violations of the Collins-Gisin-Linden-Masser-Popescu inequality.
We propose two semi-device-independent approaches that are able to quantify unknown multipartite quantum entanglement experimentally, where the only information that has to be known beforehand is quantum dimension, and the concept that plays a key role is nondegenerate Bell inequalities. Specifically, using the nondegeneracy of multipartite Bell inequalities, we obtain useful information on the purity of target quantum state. Combined with an estimate of the maximal overlap between the target state and pure product states, and a continuous property of the geometric measure of entanglement, we shall prove the information on purity allows us to give a lower bound for this entanglement measure. In addition, we show that a different combination of the above results also converts to a lower bound for the relative entropy of entanglement. As a demonstration, we apply our approach on 5-partite qubit systems with the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality, and show that useful lower bounds for the geometric measure of entanglement can be obtained if the Bell expression value is larger than 3.60, and those for the relative entropy of entanglement can be given if the Bell expression value is larger than 3.80, where the Tsirelson bound is 4.
Recently, a different form of quantum steering, i.e., certification of quantum steering in a one-sided semi-device-independent way, has been formulated [C. Jebarathinam et al., Phys. Rev. A 108, 042211 (2023)]. In this work, we use this phenomenon to provide operational simultaneous correlations in mutually unbiased bases as quantified by the measures in [S. Wu et al., Sci. Rep. 4, 4036 (2014)]. First, we show that for any bipartite state, such measure of simultaneous correlations in two mutually unbiased bases can be operationally identified as exhibiting one-sided semi-device-independent steering in a two-setting scenario. Next, we demonstrate that for two-qubit Bell-diagonal states, quantifying one-sided semi-device-independent steerability provides an operational quantification of information-theoretic quantification of simultaneous correlations in mutually unbiased bases. Then, we provide a different classification of two-qubit separable states with the above-mentioned information-theoretic quantification of simultaneous correlations in mutually unbiased bases and the quantification of one-sided semi-device-independent steerability. Finally, we invoke quantum steering ellipsoid formalism to shed intuitions on the operational characterization of simultaneous correlations in complementary bases of two-qubit states via one-sided semi-device-independent steerability. This provides us with a geometric visualization of the results.