Operational Entanglement Detection Based on $\mathrm{\Lambda }$-Moments

We introduce $\mathrm{\Lambda }$-moments with respect to any positive map $\mathrm{\Lambda }$. We show that these $\mathrm{\Lambda }$-moments can effectively characterize the entanglement of unknown quantum states without theirs prior reconstructions. Based on $\mathrm{\Lambda }$-moments necessary and sufficient separability criteria, as well as necessary optimized criteria are presented, which include the ones in [\href{https://link.aps.org/doi/10.1103/PhysRevLett.127.060504}{Phys. Rev. Lett. $\mathbf{\text{127}}$, 060504 (2021)}] as special cases. Detailed example is given to show that our criteria can detect bound entanglement that can not be identified by positive partial transpose criterion, with the explicit measurement operators to experimentally measure the corresponding $\mathrm{\Lambda }$-moments.