of E$_{R}$ ( ρ ) for the arbitrary mixture ρ we have to understand how these two properties are modified when Bloch and correlation vectors of ρ are arbitrary. The research into these issues is in progress and will be reported elsewhere. Another interesting issue, which we will go further, is to explore the REE and the dis- tillable entanglement for the higher-qubit or qudit systems. Few years ago, the analytical expressions of the distillable entanglement are obtained for some higher-dimensional bipar- tite states[16–18]. Authors in those references used the upper-bound criterion D ≤ E$_{R}$ and separability property of the Smolin’s unlockable state[15] in various cuts. We would like to modify Eq.(1.6) to be applicable not only for low-rank σ ∗ but also for higher-dimensional system. If the generalization of Eq.(1.6) is possible, we can use it to compute the REE and the distillable entanglement for many more higher-dimensional states. It may enable us to understand the nature of the optimal (or near-optimal) purification protocols. This work is in progress too. This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (2009-0073997). Acknowledgments [1] C. H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres and W. K. Wootters, Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channles , Phys. Rev. Lett. 70 (1993) 1895. [2] C. H. Bennett and S. J. Wiesner, Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states , Phys. Rev. Lett. 69 (1992) 2881. [3] V. Scarani, S. Lblisdir, N. Gisin and A. Acin, Quantum cloning , Rev. Mod. Phys. 77 (2005) 1225 [quant-ph/0511088] and references therein. [4] A. K. Ekert, Quantum Cryptography Based on Bell’s Theorem , Phys. Rev. Lett. 67 (1991) 661. [5] G. Vidal, Efficient classical simulation of slightly entangled quantum computations , Phys. Rev. Lett. 91 (2003) 147902 [quant-ph/0301063]. 21