Constrained bounds on measures of entanglement
Animesh Datta, Steven T. Flammia, Anil Shaji, Carlton M. Caves · Physical Review A · 2007
Entanglement measures constructed from two positive, but not completely positive, maps on density operators are used as constraints in placing bounds on the entanglement of formation, the tangle, and the concurrence of $4N$ mixed states. The maps are the partial transpose map and the phi map introduced by Breuer [H.-P. Breuer, Phys. Rev. Lett. 97, 080501 (2006)]. The norm-based entanglement measures constructed from these two maps, called negativity and phi negativity, respectively, lead to two sets of bounds on the entanglement of formation, the tangle, and the concurrence. We compare these bounds and identify the sets of $4N$ density operators for which the bounds from one constraint are better than the bounds from the other. In the process, we present a derivation of the already known bound on the concurrence based on the negativity. We compute bounds on the three measures of entanglement using both the constraints simultaneously. We demonstrate how such doubly constrained bounds can be constructed. We discuss extensions of our results to bipartite states of higher dimensions and with more than two constraints.