Characteristics and benchmarks of entanglement of mixed states: The two-qubit case

Abstract

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