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The dynamic behaviors of local quantum uncertainty for three-qubit X states under decoherence channels

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Abstract

We derive the analytical expression of local quantum uncertainty for three-qubit X states. We give also the expressions of quantum discord and the negativity. A comparison of these three quantum correlations quantifiers is discussed in the special cases of mixed GHZ states and Bell-type states. We find that local quantum uncertainty gives the same amount of non-classical correlations as are measured by entropic quantum discord and goes beyond negativity. We also discuss the dynamics of non-classical correlations under the effect of phase damping, depolarizing and phase reversal channels. We find the local quantum uncertainty shows more robustness and exhibits, under phase reversal effect, revival and frozen phenomena. The monogamy property of local quantum uncertainty is also discussed. It is shown that it is monogamous for three-qubit states.

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Author information

Authors and Affiliations

  1. LPHE-Modeling and Simulation, Faculty of Sciences, University Mohammed V, Rabat, Morocco

    A. Slaoui & R. Ahl Laamara

  2. Department of Physics, Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco

    M. Daoud

  3. Abdus Salam International Centre for Theoretical Physics, Miramare, Trieste, Italy

    M. Daoud

  4. Centre of Physics and Mathematics, CPM, CNESTEN, Rabat, Morocco

    R. Ahl Laamara

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  1. A. Slaoui
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  2. M. Daoud
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  3. R. Ahl Laamara
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Correspondence to A. Slaoui.

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Appendix

Appendix

In this appendix, we give the necessary tools to compute the local quantum uncertainty for three-qubit X states. First, the eigenvalues corresponding to density matrix \({\rho _{123}}\) of the form (5) are given by

$$\begin{aligned} \begin{gathered} \lambda _1^ \pm = \frac{1}{2}{t_1} \pm \frac{1}{2}\sqrt{{t_1}^2 - 4{d_1}}, \\ \lambda _2^ \pm = \frac{1}{2}{t_2} \pm \frac{1}{2}\sqrt{{t_2}^2 - 4{d_2}}, \\ \lambda _3^ \pm = \frac{1}{2}{t_3} \pm \frac{1}{2}\sqrt{{t_3}^2 - 4{d_3}}, \\ \lambda _4^ \pm = \frac{1}{2}{t_4} \pm \frac{1}{2}\sqrt{{t_4}^2 - 4{d_4}}, \\ \end{gathered} \end{aligned}$$
(93)

where

$$\begin{aligned} \begin{array}{l} t_1 = \rho _{11} + \rho _{88},\\ t_2 = \rho _{22} + \rho _{77},\\ t_3 = \rho _{33} + \rho _{66},\\ t_4 = \rho _{44} + \rho _{55}, \end{array} \begin{array}{l} d_1 = \rho _{11}\rho _{88} - \rho _{18}\rho _{81},\\ d_2 = \rho _{22}\rho _{77} - \rho _{27}\rho _{72},\\ d_3 = \rho _{33}\rho _{66} - \rho _{36}\rho _{63},\\ d_4 = \rho _{44}\rho _{55} - \rho _{45}\rho _{54}. \end{array} \end{aligned}$$
(94)

The matrix \(\sqrt{{\rho _{123}}}\) is also X-shaped and has the form

$$\begin{aligned} \sqrt{{\rho _{123}}} = \left( {\begin{array}{*{20}{c}} {\frac{{{\rho _{11}} + \sqrt{{d_1}} }}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }}}&{}0&{}0&{}0&{}0&{}0&{}0&{}{\frac{{{\rho _{18}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }}} \\ 0&{}{\frac{{{\rho _{22}} + \sqrt{{d_2}} }}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }}}&{}0&{}0&{}0&{}0&{}{\frac{{{\rho _{27}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }}}&{}0 \\ 0&{}0&{}{\frac{{{\rho _{33}} + \sqrt{{d_3}} }}{{\sqrt{{t_3} + 2\sqrt{{d_3}} } }}}&{}0&{}0&{}{\frac{{{\rho _{36}}}}{{\sqrt{{t_3} + 2\sqrt{{d_3}} } }}}&{}0&{}0 \\ 0&{}0&{}0&{}{\frac{{{\rho _{44}} + \sqrt{{d_4}} }}{{\sqrt{{t_4} + 2\sqrt{{d_4}} } }}}&{}{\frac{{{\rho _{45}}}}{{\sqrt{{t_4} + 2\sqrt{{d_4}} } }}}&{}0&{}0&{}0 \\ 0&{}0&{}0&{}{\frac{{{\rho _{54}}}}{{\sqrt{{t_4} + 2\sqrt{{d_4}} } }}}&{}{\frac{{{\rho _{55}} + \sqrt{{d_4}} }}{{\sqrt{{t_4} + 2\sqrt{{d_4}} } }}}&{}0&{}0&{}0 \\ 0&{}0&{}{\frac{{{\rho _{63}}}}{{\sqrt{{t_3} + 2\sqrt{{d_3}} } }}}&{}0&{}0&{}{\frac{{{\rho _{66}} + \sqrt{{d_3}} }}{{\sqrt{{t_3} + 2\sqrt{{d_3}} } }}}&{}0&{}0 \\ 0&{}{\frac{{{\rho _{72}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }}}&{}0&{}0&{}0&{}0&{}{\frac{{{\rho _{77}} + \sqrt{{d_2}} }}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }}}&{}0 \\ {\frac{{{\rho _{81}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }}}&{}0&{}0&{}0&{}0&{}0&{}0&{}{\frac{{{\rho _{88}} + \sqrt{{d_1}} }}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }}} \end{array}} \right) . \end{aligned}$$

In the Fano–Bloch representation, the matrix \(\sqrt{{\rho _{123}}}\) rewrites as

$$\begin{aligned} \sqrt{{\rho _{123}}} = \sum \limits _{\chi \delta \eta } {{T_{\chi \delta \eta }}} {\sigma _\chi } \otimes {\sigma _\delta } \otimes {\sigma _\eta }, \end{aligned}$$
(95)

where \(\chi ,\delta ,\eta = 0,1,2,3\) and the Fano–Bloch parameters are defined by \({T_{\chi \delta \eta }} = \mathrm{tr}\left( {\sqrt{{\rho _{123}}} {\sigma _\chi } \otimes {\sigma _\delta } \otimes {\sigma _\eta }} \right) \).

The non-vanishing elements \({T_{\chi \delta \eta }}\) are given by

$$\begin{aligned} {T_{111}}&= \frac{{{R_{111}} - {R_{221}} - {R_{122}} - {R_{212}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{111}} - {R_{221}} + {R_{122}} + {R_{212}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{111}} + {R_{221}} + {R_{122}} - {R_{212}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{111}} + {R_{221}} - {R_{122}} + {R_{212}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(96)
$$\begin{aligned} {T_{211}}&= \frac{{{R_{112}} + {R_{121}} + {R_{211}} - {R_{222}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{222}} + {R_{121}} + {R_{211}} - {R_{112}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{121}} - {R_{211}} - {R_{112}} - {R_{222}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{121}} + {R_{112}} + {R_{222}} - {R_{211}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(97)
$$\begin{aligned} {T_{121}}&= \frac{{{R_{112}} + {R_{121}} + {R_{211}} - {R_{222}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{112}} - {R_{222}} - {R_{121}} - {R_{211}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{211}} + {R_{112}} + {R_{222}} - {R_{121}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{121}} + {R_{112}} + {R_{222}} - {R_{211}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(98)
$$\begin{aligned} {T_{221}}&= \frac{{{R_{221}} + {R_{122}} + {R_{212}} - {R_{111}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{111}} - {R_{221}} + {R_{122}} + {R_{212}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{111}} + {R_{221}} + {R_{122}} - {R_{212}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{122}} - {R_{111}} - {R_{221}} - {R_{212}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(99)
$$\begin{aligned} {T_{112}}&= \frac{{{R_{112}} + {R_{121}} + {R_{211}} - {R_{222}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{222}} + {R_{121}} + {R_{211}} - {R_{112}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{211}} + {R_{112}} + {R_{222}} - {R_{121}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{211}} - {R_{121}} - {R_{112}} - {R_{222}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(100)
$$\begin{aligned} {T_{122}}&= \frac{{{R_{221}} + {R_{122}} + {R_{212}} - {R_{111}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{111}} - {R_{221}} + {R_{122}} + {R_{212}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{212}} - {R_{111}} - {R_{221}} - {R_{122}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{111}} + {R_{221}} - {R_{122}} + {R_{212}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(101)
$$\begin{aligned} {T_{212}}&= \frac{{{R_{221}} + {R_{122}} + {R_{212}} - {R_{111}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{221}} - {R_{111}} - {R_{122}} - {R_{212}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{111}} + {R_{221}} + {R_{122}} - {R_{212}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{111}} + {R_{221}} - {R_{122}} + {R_{212}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(102)
$$\begin{aligned} {T_{222}}&= \frac{{{R_{222}} - {R_{112}} - {R_{121}} - {R_{211}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{222}} + {R_{121}} + {R_{211}} - {R_{112}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{211}} + {R_{112}} + {R_{222}} - {R_{121}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{121}} + {R_{112}} + {R_{222}} - {R_{211}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(103)
$$\begin{aligned} {T_{000}}&= \sqrt{{t_1} + 2\sqrt{{d_1}} } + \sqrt{{t_2} + 2\sqrt{{d_2}} } + \sqrt{{t_3} + 2\sqrt{{d_3}} } + \sqrt{{t_4} + 2\sqrt{{d_4}} }, \end{aligned}$$
(104)
$$\begin{aligned} {T_{030}}&= \frac{{{R_{030}} + {R_{300}} + {R_{003}} + {R_{333}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{333}} + {R_{003}} - {R_{030}} - {R_{300}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{300}} + {R_{003}} - {R_{030}} - {R_{333}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{030}} - {R_{300}} + {R_{003}} - {R_{333}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(105)
$$\begin{aligned} {T_{300}}&= \frac{{{R_{030}} + {R_{300}} + {R_{003}} + {R_{333}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} - \frac{{{R_{333}} + {R_{003}} - {R_{030}} - {R_{300}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad - \frac{{{R_{300}} + {R_{003}} - {R_{030}} - {R_{333}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{030}} - {R_{300}} + {R_{003}} - {R_{333}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(106)
$$\begin{aligned} {T_{330}}&= \sqrt{{t_1} + 2\sqrt{{d_1}} } - \sqrt{{t_2} + 2\sqrt{{d_2}} } - \sqrt{{t_3} + 2\sqrt{{d_3}} } + \sqrt{{t_4} + 2\sqrt{{d_4}} }, \end{aligned}$$
(107)
$$\begin{aligned} {T_{003}}&= \frac{{{R_{030}} + {R_{300}} + {R_{003}} + {R_{333}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{030}} + {R_{300}} - {R_{333}} - {R_{003}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{300}} + {R_{003}} - {R_{030}} - {R_{333}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{300}} + {R_{333}} - {R_{003}} - {R_{030}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}, \end{aligned}$$
(108)
$$\begin{aligned} {T_{033}}&= \sqrt{{t_1} + 2\sqrt{{d_1}} } - \sqrt{{t_2} + 2\sqrt{{d_2}} } + \sqrt{{t_3} + 2\sqrt{{d_3}} } - \sqrt{{t_4} + 2\sqrt{{d_4}} }, \end{aligned}$$
(109)
$$\begin{aligned} {T_{303}}&= \sqrt{{t_1} + 2\sqrt{{d_1}} } + \sqrt{{t_2} + 2\sqrt{{d_2}} } - \sqrt{{t_3} + 2\sqrt{{d_3}} } - \sqrt{{t_4} + 2\sqrt{{d_4}} }, \end{aligned}$$
(110)
$$\begin{aligned} {T_{333}}&= \frac{{{R_{030}} + {R_{300}} + {R_{003}} + {R_{333}}}}{{4\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{{{R_{333}} + {R_{003}} - {R_{030}} - {R_{300}}}}{{4\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \nonumber \\&\quad + \frac{{{R_{030}} + {R_{333}} - {R_{300}} - {R_{003}}}}{{4\sqrt{{t_3} + 2\sqrt{{d_3}} } }} + \frac{{{R_{300}} + {R_{333}} - {R_{003}} - {R_{030}}}}{{4\sqrt{{t_4} + 2\sqrt{{d_4}} } }}. \end{aligned}$$
(111)

The eigenvalues \(\sqrt{\lambda _1^ \pm } \),\(\sqrt{\lambda _2^ \pm }\), \(\sqrt{\lambda _3^ \pm } \) and \(\sqrt{\lambda _4^ \pm } \) of the matrix \(\sqrt{{\rho _{123}}} \) are given by

$$\begin{aligned} \begin{array}{l} \sqrt{\lambda _1^ \pm } = \frac{1}{2}\sqrt{{t_1} + 2\sqrt{{d_1}} } \pm \sqrt{{t_1} - 2\sqrt{{d_1}} }, \\ \sqrt{\lambda _2^ \pm } = \frac{1}{2}\sqrt{{t_2} + 2\sqrt{{d_2}} } \pm \sqrt{{t_2} - 2\sqrt{{d_2}} }, \\ \sqrt{\lambda _3^ \pm } = \frac{1}{2}\sqrt{{t_3} + 2\sqrt{{d_3}} } \pm \sqrt{{t_3} - 2\sqrt{{d_3}} }, \\ \sqrt{\lambda _4^ \pm } = \frac{1}{2}\sqrt{{t_4} + 2\sqrt{{d_4}} } \pm \sqrt{{t_4} - 2\sqrt{{d_4}} }. \end{array} \end{aligned}$$
(112)

The elements of the matrix W defined by (3) are given in terms of \({R_{\alpha \beta \gamma }}\) and \(\sqrt{\lambda _{'i}^ \pm } \) by

$$\begin{aligned} {w_{12}}&= {w_{21}} = \frac{{\left( {{R_{111}} - {R_{212}}} \right) \left( {{R_{121}} - {R_{222}}} \right) + \left( {{R_{221}} + {R_{122}}} \right) \left( {{R_{112}} + {R_{211}}} \right) }}{{8\left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) \left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) }} \\&\quad + \frac{{\left( {{R_{111}} + {R_{212}}} \right) \left( {{R_{121}} + {R_{222}}} \right) + \left( {{R_{122}} - {R_{221}}} \right) \left( {{R_{112}} - {R_{211}}} \right) }}{{8\left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) \left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) }}\\ {w_{11}}&= \left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) \left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) + \left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) \left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) \\&\quad +\frac{{{{\left( {{R_{300}} + {R_{003}}} \right) }^2} + {{\left( {{R_{111}} - {R_{212}}} \right) }^2} - {{\left( {{R_{221}} + {R_{122}}} \right) }^2} + {{\left( {{R_{112}} + {R_{211}}} \right) }^2} - {{\left( {{R_{121}} - {R_{222}}} \right) }^2} - {{\left( {{R_{030}} + {R_{333}}} \right) }^2}}}{{16\left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) \left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) }} \\&\quad +\frac{{{{\left( {{R_{111}} + {R_{212}}} \right) }^2} - {{\left( {{R_{122}} - {R_{221}}} \right) }^2} + {{\left( {{R_{112}} - {R_{211}}} \right) }^2} - {{\left( {{R_{121}} + {R_{222}}} \right) }^2} + {{\left( {{R_{003}} - {R_{300}}} \right) }^2} - {{\left( {{R_{333}} - {R_{030}}} \right) }^2}}}{{16\left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) \left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) }},\\ {w_{22}}&= \left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) \left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) + \left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) \left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) \\&\quad + \frac{{{{\left( {{R_{221}} + {R_{122}}} \right) }^2} - {{\left( {{R_{111}} - {R_{212}}} \right) }^2} + {{\left( {{R_{121}} - {R_{222}}} \right) }^2} - {{\left( {{R_{112}} + {R_{211}}} \right) }^2} + {{\left( {{R_{300}} + {R_{003}}} \right) }^2} - {{\left( {{R_{030}} + {R_{333}}} \right) }^2}}}{{16\left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) \left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) }} \\&\quad +\frac{{{{\left( {{R_{122}} - {R_{221}}} \right) }^2} - {{\left( {{R_{111}} + {R_{212}}} \right) }^2} + {{\left( {{R_{121}} + {R_{222}}} \right) }^2} - {{\left( {{R_{112}} - {R_{211}}} \right) }^2} + {{\left( {{R_{003}} - {R_{300}}} \right) }^2} - {{\left( {{R_{333}} - {R_{030}}} \right) }^2}}}{{16\left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) \left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) }},\\ {w_{33}}&=\frac{1}{2}\left[ {{{\left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) }^2} + {{\left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) }^2} + {{\left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) }^2} + {{\left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) }^2}} \right] \\&\quad + \frac{1}{{32}}\left[ {\frac{{{{\left( {{R_{033}} + {R_{300}} + {R_{003}} + {R_{333}}} \right) }^2} - {{\left( {{R_{112}} + {R_{121}} + {R_{211}} - {R_{222}}} \right) }^2} - {{\left( {{R_{111}} - {R_{221}} - {R_{122}} - {R_{212}}} \right) }^2}}}{{{{\left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) }^2}}}} \right] \\&\quad + \frac{1}{{32}}\left[ {\frac{{{{\left( {{R_{333}} + {R_{003}} - {R_{300}} - {R_{030}}} \right) }^2} - {{\left( {{R_{222}} + {R_{121}} + {R_{211}} - {R_{112}}} \right) }^2} - {{\left( {{R_{111}} - {R_{221}} + {R_{122}} + {R_{212}}} \right) }^2}}}{{{{\left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) }^2}}}} \right] \\&\quad + \frac{1}{{32}}\left[ {\frac{{{{\left( {{R_{300}} + {R_{003}} - {R_{030}} - {R_{333}}} \right) }^2} - {{\left( {{R_{121}} - {R_{211}} - {R_{112}} - {R_{222}}} \right) }^2} - {{\left( {{R_{111}} + {R_{221}} + {R_{122}} - {R_{212}}} \right) }^2}}}{{{{\left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) }^2}}}} \right] \\&\quad + \frac{1}{{32}}\left[ {\frac{{{{\left( {{R_{030}} - {R_{300}} + {R_{003}} - {R_{333}}} \right) }^2} - {{\left( {{R_{121}} - {R_{211}} + {R_{112}} + {R_{222}}} \right) }^2} - {{\left( {{R_{111}} + {R_{221}} - {R_{122}} + {R_{212}}} \right) }^2}}}{{{{\left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) }^2}}}} \right] . \end{aligned}$$

After some long but feasible calculations, we get

$$\begin{aligned} {w_{12}}&= {w_{21}} = \frac{{2i\left( {{\rho _{18}}{\rho _{72}} - {\rho _{81}}{\rho _{27}}} \right) }}{{\left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) \left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) }}\nonumber \\&\quad + \frac{{2i\left( {{\rho _{36}}{\rho _{54}} - {\rho _{45}}{\rho _{63}}} \right) }}{{\left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) \left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) }}, \end{aligned}$$
(113)
$$\begin{aligned} {w_{11}}&= \left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) \left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) + \left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) \left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) \nonumber \\&\quad + \frac{{\left( {{\rho _{11}} - {\rho _{88}}} \right) \left( {{\rho _{22}} - {\rho _{77}}} \right) + 2\left( {{\rho _{18}}{\rho _{72}} + {\rho _{81}}{\rho _{27}}} \right) }}{{\left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) \left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) }} \nonumber \\&\quad + \frac{{\left( {{\rho _{44}} - {\rho _{55}}} \right) \left( {{\rho _{33}} - {\rho _{66}}} \right) + 2\left( {{\rho _{36}}{\rho _{54}} + {\rho _{45}}{\rho _{63}}} \right) }}{{\left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) \left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) }}, \end{aligned}$$
(114)
$$\begin{aligned} {w_{22}}&= \left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) \left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) + \left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) \left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) \nonumber \\&\quad + \frac{{\left( {{\rho _{11}} - {\rho _{88}}} \right) \left( {{\rho _{22}} - {\rho _{77}}} \right) - 2\left( {{\rho _{18}}{\rho _{72}} + {\rho _{81}}{\rho _{27}}} \right) }}{{\left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) \left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) }} \nonumber \\&\quad + \frac{{\left( {{\rho _{44}} - {\rho _{55}}} \right) \left( {{\rho _{33}} - {\rho _{66}}} \right) - 2\left( {{\rho _{36}}{\rho _{54}} + {\rho _{45}}{\rho _{63}}} \right) }}{{\left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) \left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) }}, \end{aligned}$$
(115)
$$\begin{aligned} {w_{33}}&= \frac{1}{2}\left( {1 + 2\sum \limits _{i = 1}^4 {\sqrt{{d_i}} } } \right) + \frac{{{{\left( {2{\rho _{11}} + {\rho _{66}} - {\rho _{88}} - {\rho _{77}} - {\rho _{55}}} \right) }^2} - 16{\rho _{18}}{\rho _{81}}}}{{8{{\left( {\sqrt{\lambda _1^ + } + \sqrt{\lambda _1^ - } } \right) }^2}}}\nonumber \\&\quad + \frac{{{{\left( {{\rho _{33}} - {\rho _{66}}} \right) }^2} - 4{\rho _{36}}{\rho _{63}}}}{{2{{\left( {\sqrt{\lambda _4^ + } + \sqrt{\lambda _4^ - } } \right) }^2}}} \nonumber \\&\quad + \frac{{{{\left( {{\rho _{44}} - {\rho _{55}}} \right) }^2} + {{\left( {{\rho _{63}} - {\rho _{36}}} \right) }^2} - {{\left( {{\rho _{45}} + {\rho _{54}}} \right) }^2}}}{{2{{\left( {\sqrt{\lambda _2^ + } + \sqrt{\lambda _2^ - } } \right) }^2}}} \nonumber \\&\quad + \frac{{{{\left( {{\rho _{22}} - {\rho _{77}}} \right) }^2} - 4{\rho _{27}}{\rho _{72}}}}{{2{{\left( {\sqrt{\lambda _3^ + } + \sqrt{\lambda _3^ - } } \right) }^2}}}. \end{aligned}$$
(116)

with \(\lambda ^{\pm } (i=1,2, 3, 4)\) are the eigenvalues of the density matrix \(\rho _{123}\) (5).

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Slaoui, A., Daoud, M. & Ahl Laamara, R. The dynamic behaviors of local quantum uncertainty for three-qubit X states under decoherence channels. Quantum Inf Process 18, 250 (2019). https://doi.org/10.1007/s11128-019-2363-x

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