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

TL;DR

This paper introduces a probability density function to characterize entanglement in two-qubit mixed states, providing a more comprehensive measure than traditional benchmarks like concurrence and negativity.

Contribution

It constructs an explicit entanglement distribution function for two-qubit states, invariant under local transformations, and capable of reconstructing the state with minimal additional parameters.

Findings

The new measure offers a detailed entanglement profile for two-qubit states.

It resolves debates about entanglement's role in NMR quantum computing.

The measure is invariant under local unitary operations.

Abstract

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 \textit{concurrence} and \textit{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.