Constructive entanglement test from triangle inequality
  We derive a simple lower bound on the geometric measure of entanglement for mixed quantum states in the case of a general multipartite system. The main ingredient of the presented derivation is the triangle inequality applied to the root infidelity distance in the space of density matrices. The obtained bound leads to entanglement criteria with a straightforward interpretation. Proposed criteria provide an experimentally accessible, powerful entanglement test. 
{\L}ukasz Rudnicki, Zbigniew Pucha{\l}a, Pawe{\l} Horodecki and Karol   Zyczkowski
quant-ph