It is known that the witness operator is useful in the detection and quantification of entangled states. This motivated us for the construction of the family of witness operators that can detect many mixed entangled states. This family of witness operators is then used to estimate the lower bound of concurrence of the detected mixed entangled states. Our method of construction of witness operator is important in the sense that it will estimate a better lower bound of concurrence of the entangled states in arbitrary \(d_{1}\otimes d_{2} (d_{1}\le d_{2})\)-dimensional system compared to the lower bound of the concurrence given in Chen et al. (Phys Rev Lett 95:040504, 2005). We have shown the significance of our constructed witness operator by detecting many bound entangled states that are not detected by the earlier methods, and then, we use the expectation value of the witness operator to estimate the lower bound of the concurrence of those bound entangled states.

已知证人操作员在纠缠状态的检测和量化中很有用。这促使我们构建可以检测许多混合纠缠状态的证人操作员家庭。然后使用该证人算子家族来估计检测到的混合纠缠状态的并发下限。我们的见证运算符的构造方法很重要，因为它可以估计任意\（d_ {1} \ otimes d_ {2}（d_ {1} \ le d_ {2 }）\）维系统，与Chen等人给出的并发下限相比。（Phys Rev Lett 95：040504，2005）。通过检测许多早期方法无法检测到的绑定纠缠态，我们已经证明了构造的见证算子的重要性，然后，我们使用见证算子的期望值来估计那些绑定纠缠态并发性的下界。