4 8 10 12 14 16 18 20 N$_{p}$ 0.000 0.001 0.002 0.003 0.004 0.005 p=0.65 p=0.7 p=0.75 p=0.8 p=0.9 0. In practice, we have calculated τ 3 up to N$_{p}$ =20. We find that use N$_{p}$ =15 can converge τ 3 to less than 10 $_{−}$3 for this problem. (See Fig. 2). The calculated results are shown in Fig.1(d). From Fig. 1(d), we see that τ 3 of the white noise mixed GHZ states is zero for p less than about 0.7, then increase almost linearly to 1.0 as p increases. The behavior of τ 3 looks similar to that of mixed GHZ and W states. It is straight forward to apply the method to any mixed states. It therefore opens up a way to study the multi- particle entanglement effects in many important systems, e.g., in systems with quantum phase transitions, which was impossible before, because the analytical solution to the tree-tangle of a general mixed state is not available. FIG. 2: (Color online) The N$_{p}$ dependent Δ τ 3 calculated with replica exchange for density matrix constructed by GHZ state with white noise, where Δ τ 3 = τ 3 − τ min 3 with τ min 3 the min- imum three-tangle for each p value. To conclude, we have demonstrate an efficient numeri- cal method to calculate the three-tangle of general mixed states. We construct a “energy function” (target func- tion) for the three-tangle of the mixed state under cer- tain constrains. The “energy function” (target function) is then optimized via a replica exchange Monte Carlo method. We have tested the method for the examples with known analytical results, showing remarkable agree- ment. We further calculate the three-tangle of GHZ states mixed with white noise. The work opens a way to study three-tangle of general mixed states, and can be generalized to solve many other optimization problems in quantum information theory. accuracy can easily be improved by increasing κ and de- creasing T min . For the example given above, in the case of p=0.8, if we reduce T min to 10 $_{−}$10 and use κ = 10 $^{10}$, we obtain τ 3 =0.18349906 compared to the analytical value 0.18349861 in an accuracy ∼ 10 $^{$_{−}$7}$. In principle, our nu- merical method can achieve arbitrary accuracy. We have also tried to optimize Eq. 13 via traditional CSA method. We find the performance of the traditional method is very bad, because it can easily be trapped at some local min- ima. Having demonstrated the ability of the method, we then calculate the τ 3 of the GHZ states mixed with white noise, which does not have analytical solutions so far. The density matrices of the states are L.H. acknowledges the support from the Chinese Na- tional Fundamental Research Program 2006CB921900, the Innovation funds and “Hundreds of Talents” program from Chinese Academy of Sciences, and National Natural Science Foundation of China, Grant No. 10674124. W.Z were supported by National Natural Science Foundation of China, Grant No. 10874170 and K.C.Wong Education Foundation, Hong Kong. where π$_{E}$ is a unit matrix. In this case, the density matri- ces are of rank 8. We need at least N$_{p}$ =8 to ensure R 2 ∼ ρ ( p ) = p π$_{GHZ}$ + 1 − p 8 π$_{E}$ , (24) [1] C. H. Bennett, D. DiVincenzo, J. Smolin, and W. K. Wootters, Phys. Rev. A 54 , 3824 (1996). [2] V. Coffman, J. Kundu, and W. Wootters, Phys. Rev. A 61 , 052306 (2000). [3] Y.-K. Bai, D. Yang, and Z.-D. Wang, Phys. Rev. A 76 , 022336 (2007). [4] J.-M. Cai, Z.-W. Zhou, X.-X. Zhou, and G.-C. Guo, Phys. Rev. A 74 , 042338 (2006). [5] A. Uhlmann, Phys. Rev. A 62 , 032307 (2000). [6] W. K. Wooters, Phys. Rev. Lett. 80 , 2245 (1998). [7] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80 , 518 (2008). [8] R. Lohmayer, A. Osterloh, J. Siewert, and A. Uhlmann, Phys. Rev. Lett. 97 , 260502 (2006). [9] C. Eltschka, A. Osterloh, J. Siewert, and A. Uhlmann, arXiv:0711.4477v1 (2007). [10] E. Jung, M.-R. Hwang, D. Park, and J.-W. Son, Phys. Rev. A 79 , 024306 (2009). [11] E. Jung, D. Park, and J.-W. Son, Phys. Rev. A 80 , 010301(R) (2009). [12] R. H. Swendsen and J. S. Wang, Phys. Rev. Lett. 57 , 2607 (1986). [13] C. J. Geyer, Computer Science and Statistics, Proceed- ings of the 23rd Symposium on the interface (Interface Foundation, 1991). [14] E. Marinari, G. Parisi, and J. J. Ruiz-Lorenzo, Phys. Rev. B 58 , 14852 (1998). [15] Y. Sugita, A. Kitao, and Y. Okamato, J. Chem. Phys. 113 , 6042 (2000). [16] J. P. Neirotti, F. Calvo, D. L. Freeman, and J. D. Doll, J. Chem. Phys. 112 , 10340 (2000). [17] F. Benatti, H. Narnhofer, and A. Uhlmann, Rep. Math. Phys. 38 , 123 (1996). [18] E. Marinari, G. Parisi, and J. J. Ruiz-Lorenzo, in Spin Glasses and Random Fields , edited by A. P. Young (World Scientific, Singapore, 1997). [19] A. Kone and D. A. Kofke, J. Chem. Phys. 122 , 206101 (2005). [20] B. A. Berg, Markov Chain Monte Carlo Simualtions and Their Statistical Analysis (World Scientific Press, 2004).