Characteristics and benchmarks of entanglement of mixed states -- the two qubit case

We propose that the entanglement of mixed states is characterised properly in terms of a probability density function ${\mathcal{P}}_{\rho }(\mathcal{E})$. There is a need for such a measure since the prevalent measures (such as $\mathit{\text{concurrence}}$ and $\mathit{\text{negativity}}$) for two qubit systems are rough benchmarks, and not monotones of each other. Focussing on the two qubit states, we provide an explicit construction of ${\mathcal{P}}_{\rho }(\mathcal{E})$ and show that it is characterised by a set of parameters, of which concurrence is but one particular combination. ${\mathcal{P}}_{\rho }(\mathcal{E})$ is manifestly invariant under $SU(2)\times SU(2)$ transformations. It can, in fact, reconstruct the state up to local operations - with the specification of at most four additional parameters. Finally the new measure resolves the controversy regarding the role of entanglement in quantum computation in NMR systems.