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. The proposed criteria provide an experimentally accessible, powerful entanglement test. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to {\textquoteright}50 years of Bell{\textquoteright}s theorem{\textquoteright}.

}, doi = {10.1088/1751-8113/47/42/424035}, url = {https://doi.org/10.1088/1751-8113/47/42/424035}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and Karol {\.Z}yczkowski} } @article {iitisid_0589, title = {Collectibility for Mixed Quantum States}, journal = {Phys. Rev. A}, volume = {86}, number = {6}, year = {2012}, note = {arXiv:1211.0573 doi:10.1103/PhysRevA.86.062329}, pages = {062329}, abstract = {Bounds analogous to entropic uncertainty relations allow one to design practical tests to detect quantum entanglement by a collective measurement performed on several copies of the state analyzed. This approach, initially worked out for pure states only [ Phys. Rev. Lett. 107 150502 (2011)], is extended here for mixed quantum states. We define collectibility for any mixed states of a multipartite system. Deriving bounds for collectibility for positive partially transposed states of given purity provides insight into the structure of entangled quantum states. In the case of two qubits the application of complementary measurements and coincidence based detections leads to a test of entanglement of pseudopure states.}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and Karol {\.Z}yczkowski} } @article {iitisid_0421, title = {Sub- and super-fidelity as bounds for quantum fidelity}, journal = {Quantum Information \& Computation}, volume = {9}, number = {1\&2}, year = {2009}, note = {arXiv:0805.2037 IF=2.980(209); IF5=2.402(2010);}, month = {1}, pages = {0103{\textendash}0130}, abstract = {We derive several bounds on fidelity between quantum states. In particular we show that fidelity is bounded from above by a simple to compute quantity we call super{\textendash}fidelity. It is analogous to another quantity called sub{\textendash}fidelity. For any two states of a two{\textendash}dimensional quantum system ($N=2$) all three quantities coincide. We demonstrate that sub{\textendash} and super{\textendash}fidelity are concave functions. We also show that super{\textendash}fidelity is super{\textendash}multiplicative while sub{\textendash}fidelity is sub{\textendash}multiplicative and design feasible schemes to measure these quantities in an experiment. Super{\textendash}fidelity can be used to define a distance between quantum states. With respect to this metric the set of quantum states forms a part of a $N^2-1$ dimensional hypersphere.}, issn = {1533-7146}, author = {J.A. Miszczak and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and A. Uhlmann and Karol {\.Z}yczkowski} }