Entanglement Irreversibility from Quantum Discord and Quantum Deficit

We relate the problem of irreversibility of entanglement with the recently defined measures of quantum correlation—quantum discord and one-way quantum deficit. We show that the entanglement of formation is always strictly larger than the coherent information and the entanglement cost is also larger in most cases. We prove irreversibility of entanglement under local operations and classical communication for a family of entangled states. This family is a generalization of the maximally correlated states for which we also give an analytic expression for the distillable entanglement, the relative entropy of entanglement, the distillable secret key, and the quantum discord.

Figure 1

Entanglement dilution-distillation cycle. The entanglement loss is given by $\Delta$. In the case of reversible entanglement, $\Delta$ vanishes. In the irreversible case of Eqs. (8, 9), $\Delta$ is the regularized quantum discord.

Figure 2

$E^{\mathcal{C}}$, $E^{\mathcal{D}}$, and ${\Delta }_{a|c}$ for 1-MC states of 2-qubits ${\sigma }_{ab}$ with (a) $\phi =\pi /6$ and (b) $\phi =\pi /4$ in Eq. (11). The values satisfy Eq. (9), and $E^{\mathcal{D}}$ was previously known only for the case $\theta =\pi /2$.