Erratum to: Bound Entanglement for Bipartite and Tripartite Quantum Systems

Erratum to: Int J Theor Phys (2015)

DOI 10.1007/s10773-015-2563-9

The Journal has been informed of errors in the article named Bound Entanglement for Bipartite and Tripartite Quantum Systems by Hui Zhao and Sha Guo.

• (1) The error occur in Section 2, the entries $C_{k1}^{\dagger }$ and $C_{k2}^{\dagger }$ of the matrixes $\sigma _{0}=\left (\begin {array}{cccc} C_{11} & C_{12} & {\cdots } & C_{1k} \\ C_{12}^{\dagger } & C_{22}&\cdots &C_{2k}\\ {\vdots } &\vdots & {\ddots } & {\vdots } \\ C_{k1}^{\dagger }& C_{k2}^{\dagger } & {\cdots } & C_{kk} \\ \end {array} \right )$ and $\sigma =\varepsilon \left (\begin {array}{cccc} C_{11} & C_{12} & {\cdots } & C_{1k} \\ C_{12}^{\dagger } & C_{22}&\cdots &C_{2k}\\ {\vdots } &\vdots & {\ddots } & {\vdots } \\ C_{k1}^{\dagger }& C_{k2}^{\dagger } & {\cdots } & C_{kk} \\ \end {array} \right )+(1-\varepsilon )\left (\begin {array}{ccc} D_{11} & {\cdots } & 0 \\ {\vdots } & {\ddots } & {\vdots } \\ 0 & {\cdots } & D_{kk} \\ \end {array} \right )$ should be $C_{1k}^{\dagger }$ and $C_{2k}^{\dagger }$.

• (2) The error occur in Section 3, the matrix $E_{11}=\left (\begin {array}{cccccccc} E_{1} & 0 & E_{2} & 0 & E_{3} & 0 & 0&0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{2}^{\dagger } & 0 & E_{1} & 0 & 0 & 0 & E_{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{3}^{\dagger } & 0 & 0 & 0 & E_{5} & 0 & E_{6} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & E_{4}^{\dagger } & 0 & E_{6}^{\dagger } & 0 & E_{1} & 0\\ 0 & 0& 0& 0& 0& 0& 0& 0\\ \end {array} \right )$ should be $E_{11}=\left (\begin {array}{cccccccc} E_{1} & 0 & E_{2} & 0 & E_{3} & 0 & 0&0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{2}^{\dagger } & 0 & E_{7} & 0 & 0 & 0 & E_{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{3}^{\dagger } & 0 & 0 & 0 & E_{5} & 0 & E_{6} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & E_{4}^{\dagger } & 0 & E_{6}^{\dagger } & 0 & E_{8} & 0\\ 0 & 0& 0& 0& 0& 0& 0& 0\\ \end {array} \right ),\\$ where the entries E ₇ = − (P ₁₂ E ₂+E ₄ P ₂₄) and E ₈ = − (P ₂₄ E ₄+P ₃₄ E ₆).

• (3) The error occur in equation (36)

  $$\begin{array}{@{}rcl@{}}(r_{1},r_{2},r_{3},r_{4})^{t}\otimes(s_{1},0,s_{3},0,0,s_{6},0,s_{8},s_{9}, 0, s_{11},0,s_{13},0,s_{15},0)^{t} \end{array} $$

  should be

  $$\begin{array}{@{}rcl@{}}(r_{1},r_{2},r_{3},r_{4})^{t}\otimes(s_{1},0,s_{3},0,s_{5},0,s_{7},0, 0,s_{10},0,s_{12},0,s_{14},0,s_{16})^{t}.\end{array} $$

• (4) The error occur in equation (37), the vectors

  $$\begin{array}{@{}rcl@{}} (c_{1},c_{2},0,0,0,0,0,0,c_{9},c_{10},0,0,0,0,0,0)^{t}&\otimes&(d_{1},d_{2},d_{3},0)^{t},\\ (0,0,c_{3},c_{4},0,0,0,0,0,0,c_{11},c_{12},0,0,0,0)^{t}&\otimes&(d_{1},d_{2},0,d_{4})^{t},\\ (0,0,0,0,c_{5},c_{6},0,0,0,0,0,0,c_{13},c_{14},0,0)^{t}&\otimes&(d_{1},0,d_{3},d_{4})^{t},\\ (0,0,0,0,0,0,c_{7},c_{8},0,0,0,0,0,0,c_{15},c_{16})^{t}&\otimes&(0,d_{2},d_{3},d_{4})^{t}, \end{array} $$

  should be

  $$\begin{array}{@{}rcl@{}} (0,c_{2},0,0,0,0,0,0,c_{9},0,0,0,0,0,0,0)^{t}&\otimes&(d_{1},d_{2},d_{3},0)^{t},\\ (0,0,0,c_{4},0,0,0,0,0,0,c_{11},0,0,0,0,0)^{t}&\otimes&(d_{1},d_{2},0,d_{4})^{t},\\ (0,0,0,0,0,c_{6},0,0,0,0,0,0,c_{13},0,0,0)^{t}&\otimes&(d_{1},0,d_{3},d_{4})^{t},\\ (0,0,0,0,0,0,0,c_{8},0,0,0,0,0,0,c_{15},0)^{t}&\otimes&(0,d_{2},d_{3},d_{4})^{t}. \end{array} $$

The authors regret these errors