Ordering two-qubit states with concurrence and negativity

We study the ordering of two-qubit states with respect to the degree of bipartite entanglement using the Wootters concurrence—a measure of the entanglement of formation—and the negativity—a measure of the entanglement cost under the positive-partial-transpose-preserving operations. For two-qubit pure states, the negativity is the same as the concurrence. However, we demonstrate analytically on simple examples of various mixtures of Bell and separable states that the entanglement measures can impose different orderings on the states. We show which states, in general, give the maximally different predictions (i) when one of the states has the concurrence greater but the negativity smaller than those for the other state and (ii) when the states are entangled to the same degree according to one of the measures, but differently according to the other.