The dynamics of local quantum uncertainty and trace distance discord for two-qubit X states under decoherence: a comparative study

Abstract

We employ the concepts of local quantum uncertainty and geometric quantum discord based on the trace norm to investigate the environmental effects on quantum correlations of two bipartite quantum systems. The first one concerns a two-qubit system coupled with two independent bosonic reservoirs. We show that the trace discord exhibits frozen phenomenon contrarily to local quantum uncertainty. The second scenario deals with a two-level system, initially prepared in a separable state, interacting with a quantized electromagnetic radiation. Our results show that there exists an exchange of quantum correlations between the two-level system and its surrounding which is responsible for the revival phenomenon of non-classical correlations.

Appendices

Appendices

1.1 Appendix 1

The eigenvalues corresponding to this matrix $\rho$ (5) are given by

$$\begin{aligned} {{\lambda _1} = \frac{1}{2}{t_1} + \frac{1}{2}\sqrt{{t_1}^2 - 4{d_1}} }, \qquad {{\lambda _2} = \frac{1}{2}{t_2} + \frac{1}{2}\sqrt{{t_2}^2 - 4{d_2}} } \end{aligned}$$

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$$\begin{aligned} {{\lambda _3} = \frac{1}{2}{t_2} - \frac{1}{2}\sqrt{{t_2}^2 - 4{d_2}} }, {{\lambda _4} = \frac{1}{2}{t_1} - \frac{1}{2}\sqrt{{t_1}^2 - 4{d_1}} } \end{aligned}$$

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where

$$\begin{aligned} {t_1} = {\rho _{11}} + {\rho _{44}},\quad {d_1} = {\rho _{11}}{\rho _{44}} - {\rho _{14}}{\rho _{41}},\quad {t_2} = {\rho _{22}} + {\rho _{33}},\quad {d_2} = {\rho _{22}}{\rho _{33}} - {\rho _{32}}{\rho _{23}}.\nonumber \\ \end{aligned}$$

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The square root of the density matrix $\rho$ can be written in terms of the matrix elements $\rho$, in the computational basis, as follows:

$$\begin{aligned} \sqrt{\rho }= \left( {\begin{array}{*{20}{c}} {\frac{{{\rho _{11}} + \sqrt{{d_1}} }}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }}}&{}0&{}0&{}{\frac{{{\rho _{14}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }}}\\ 0&{}{\frac{{{\rho _{22}} + \sqrt{{d_2}} }}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }}}&{}{\frac{{{\rho _{23}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }}}&{}0\\ 0&{}{\frac{{{\rho _{32}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }}}&{}{\frac{{{\rho _{33}} + \sqrt{{d_2}} }}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }}}&{}0\\ {\frac{{{\rho _{41}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }}}&{}0&{}0&{}{\frac{{{\rho _{44}} + \sqrt{{d_1}} }}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }}} \end{array}} \right) . \end{aligned}$$

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The eigenvalues $\sqrt{{\lambda _1}}$, $\sqrt{{\lambda _2}}$,$\sqrt{{\lambda _3}}$ and $\sqrt{{\lambda _4}}$ of the matrix $\sqrt{\rho }$ can be rewritten as:

$$\begin{aligned} \sqrt{{\lambda _1}}= & {} \frac{1}{2}\sqrt{{t_1} + 2\sqrt{{d_1}} } + \frac{1}{2}\sqrt{{t_1} - 2\sqrt{{d_1}} } \end{aligned}$$

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$$\begin{aligned} \sqrt{{\lambda _2}}= & {} \frac{1}{2}\sqrt{{t_2} + 2\sqrt{{d_2}} } + \frac{1}{2}\sqrt{{t_2} - 2\sqrt{{d_2}}} \end{aligned}$$

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$$\begin{aligned} \sqrt{{\lambda _3}}= & {} \frac{1}{2}\sqrt{{t_2} + 2\sqrt{{d_2}} } - \frac{1}{2}\sqrt{{t_2} - 2\sqrt{{d_2}} } \end{aligned}$$

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$$\begin{aligned} \sqrt{{\lambda _4}}= & {} \frac{1}{2}\sqrt{{t_1} + 2\sqrt{{d_1}} } - \frac{1}{2}\sqrt{t_{1} - 2\sqrt{d_{1} } } \end{aligned}$$

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The matrix $\sqrt{\rho }$ is written in the Fano–Bloch representation as:

$$\begin{aligned} \sqrt{\rho }= \frac{1}{4}\sum \limits _{\chi ,\delta } {{\mathcal {R}_{\chi \delta }}} {\sigma _\chi } \otimes {\sigma _\delta }, \end{aligned}$$

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where $\chi ,\delta = 0,1,2,3$ and the Fano–Bloch parameters are defined by: ${\mathcal {R}_{\chi \delta }} = \mathrm{Tr}\left( {\sqrt{\rho }{\sigma _\chi } \otimes {\sigma _\delta }} \right)$. The vanishing correlation parameters ${\mathcal {R}_{\chi \delta }}$ are given by:

$$\begin{aligned} \mathcal {R}_{00}= & {} \sqrt{{t_1} + 2\sqrt{{d_1}} } + \sqrt{{t_2} + 2\sqrt{{d_2}} } \end{aligned}$$

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$$\begin{aligned} \mathcal {R}_{03}= & {} \frac{1}{2}\frac{{{T_{30}} + {T_{03}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }} - \frac{1}{2}\frac{{{T_{30}} - {T_{03}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \end{aligned}$$

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$$\begin{aligned} \mathcal {R}_{30}= & {} \frac{1}{2}\frac{{{T_{30}} + {T_{03}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }} + \frac{1}{2}\frac{{{T_{30}} - {T_{03}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \end{aligned}$$

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$$\begin{aligned} \mathcal {R}_{11}= & {} \frac{1}{2}\frac{{{T_{11}} + {T_{22}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }} + \frac{1}{2}\frac{{{T_{11}} - {T_{22}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }} \end{aligned}$$

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$$\begin{aligned} \mathcal {R}_{12}= & {} \frac{1}{2}\frac{{{T_{12}} - {T_{21}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }} + \frac{1}{2}\frac{{{T_{12}} + {T_{21}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }} \end{aligned}$$

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$$\begin{aligned} \mathcal {R}_{12}= & {} \frac{1}{2}\frac{{{T_{12}} - {T_{21}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }} + \frac{1}{2}\frac{{{T_{12}} + {T_{21}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }} \end{aligned}$$

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$$\begin{aligned} \mathcal {R}_{21}= & {} \frac{1}{2}\frac{{{T_{12}} + {T_{21}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }} - \frac{1}{2}\frac{{{T_{12}} - {T_{21}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }} \end{aligned}$$

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$$\begin{aligned} \mathcal {R}_{22}= & {} \frac{1}{2}\frac{{{T_{11}} + {T_{22}}}}{{\sqrt{{t_2} + 2\sqrt{{d_2}} } }} - \frac{1}{2}\frac{{{T_{11}} - {T_{22}}}}{{\sqrt{{t_1} + 2\sqrt{{d_1}} } }} \end{aligned}$$

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$$\begin{aligned} \mathcal {R}_{33}= & {} \sqrt{{t_1} + 2\sqrt{{d_1}} } - \sqrt{{t_2} + 2\sqrt{{d_2}} } . \end{aligned}$$

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1.2 Appendix 2

In this appendix, we give the expressions of the $\gamma \left( t \right)$ function for sub-ohmic, ohmic and super-ohmic reservoirs [30].

1.2.1 (i) Sub-ohmic reservoirs

The sub-ohmic regime corresponds to the situation where $0< s < 1$. In this case, the function $\gamma \left( t \right)$ reads as [30]

$$\begin{aligned}&\gamma \left( t \right) = \frac{{2\lambda \varGamma \left( s \right) }}{{s - 1}}\left\{ {1 - {{\left( {1 - {\varOmega ^2}{t^2}} \right) }^{{{{\left( {1 - s} \right) }} /\!\ {2}}}}\cos \left[ {\left( {s - 1} \right) \arctan \left( {\varOmega t} \right) } \right] } \right\} \\&\quad + \frac{{4\lambda \varGamma \left( s \right) }}{{s - 1}}{\sum \limits _{m = 1}^\infty {\left( {1 + m\varOmega \beta } \right) } ^{1 - s}}\\&\quad \times \left\{ {1 - {{\left[ {1 + {{\left( {\frac{{\varOmega t}}{{1 + m\varOmega \beta }}} \right) }^2}} \right] }^{{{{\left( {1 - s} \right) }} \!/ \!{2}}}}\cos \left[ {\left( {s - 1} \right) \arctan \left( {\frac{{\varOmega t}}{{1 + m\varOmega \beta }}} \right) } \right] } \right\} , \end{aligned}$$

where ${\varGamma \left( s \right) }$ denotes the Gamma function. To study the evolution of the local quantum uncertainty, one should calculate the time derivative of the function $\gamma \left( t \right)$ . It is given by

$$\begin{aligned}&{\frac{{\mathrm{d}\gamma \left( t \right) }}{{\mathrm{d}t}}= 2\lambda \varGamma \left( s \right) \varOmega } \nonumber \\&\quad {\left\{ {{{\left( {1 + {\varOmega ^2}{t^2}} \right) }^{{{ - s} / 2}}}\sin \left[ {s\arctan \left( {\varOmega t} \right) } \right] + 2{{\sum \limits _{m = 1}^\infty {\left[ {{{\left( {1 + m\varOmega \beta } \right) }^2}+ {\varOmega ^2}{t^2}} \right] } }^{{{ - s} / 2}}}\sin \left[ {s\arctan \left( {\frac{{\varOmega t}}{{1 + m\varOmega \beta }}} \right) } \right] } \right\} }\nonumber \\ \end{aligned}$$

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1.2.2 (ii) Ohmic reservoirs

For reservoirs of ohmic type $(s = 1)$, the low-frequency behavior of the function $J\left( \omega \right)$ is linear in terms of $\omega$. The Eq. (29) becomes [30]

$$\begin{aligned} \gamma \left( t \right) = \lambda \left[ {\ln \left( {1 + {\varOmega ^2}{t^2}} \right) + 4\ln \varGamma \left( {1 + \frac{1}{{\varOmega \beta }}} \right) - 2\ln {{\left| {\varGamma \left( {1 + \frac{1}{{\varOmega \beta }} + i\frac{t}{\beta }} \right) } \right| }^2}} \right] \nonumber \\ \end{aligned}$$

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The time derivative of the function $\gamma \left( t \right)$ is given by

$$\begin{aligned} \frac{{\mathrm{d}\gamma \left( t \right) }}{{\mathrm{d}t}} = 2\lambda {\varOmega ^2}t\left[ {\frac{1}{{1 + {\varOmega ^2}{t^2}}} + \sum \limits _{m = 1}^\infty {\frac{2}{{{{\left( {1 + m\varOmega \beta } \right) }^2} + {\varOmega ^2}{t^2}}}} } \right] . \end{aligned}$$

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1.2.3 (iii) Super-ohmic reservoirs

For super-ohmic reservoirs $(s > 1)$, the function $\gamma \left( t \right)$ reads as [30]

$$\begin{aligned}&\gamma \left( t \right) = 2\lambda \varGamma \left( {s - 1} \right) \left\{ {1 - {{\left( {1 + {\varOmega ^2}{t^2}} \right) }^{{{1 - s} / 2}}}\cos \left[ {\left( {s - 1} \right) \arctan \left( {\varOmega t} \right) } \right] } \right\} \\&\quad + 4\lambda \varGamma \left( {s - 1} \right) \sum \limits _{m = 1}^\infty {{{\left( {1 + m\varOmega \beta } \right) }^{1 - s}}}\\&\quad {\left\{ {1 - {{\left[ {1 + {{\left( {\frac{{\varOmega t}}{{1 + m\varOmega \beta }}} \right) }^2}} \right] }^{{{\left( {1 - s} \right) } / 2}}}\cos \left[ {\left( {s - 1} \right) \arctan \left( {\frac{{\varOmega t}}{{1 + m\varOmega \beta }}} \right) } \right] } \right\} }, \end{aligned}$$

and the derivative of $\gamma \left( t \right)$ with respect to time t is given by

$$\begin{aligned} \frac{{\mathrm{d}\gamma \left( t \right) }}{{\mathrm{d}t}}&= 2\lambda \varOmega \varGamma \left( s \right) {\left( {1 + {\varOmega ^2}{t^2}} \right) ^{{{ - s} / 2}}}\sin \left[ {s\arctan \left( {\varOmega t} \right) } \right] \\&\quad + 4\lambda \varOmega \varGamma \left( s \right) {\sum \limits _{m = 1}^\infty {\left[ {{{\left( {1 + m\varOmega \beta } \right) }^2} + {\varOmega ^2}{t^2}} \right] } ^{ - {s / 2}}}\sin \left[ {s\arctan \left( {\frac{{\varOmega t}}{{1 + m\varOmega \beta }}} \right) } \right] . \end{aligned}$$

Due to the complicated expression of $\gamma \left( t \right)$ and to simplify our numerical calculations, we have chosen the situation where $1 < s \le 2$ and the temperature equals zero. In this case case, the function $\gamma \left( t \right)$ becomes

$$\begin{aligned} \gamma \left( t \right) = 2\lambda \varGamma \left( {s - 1} \right) \left\{ {1 - {{\left( {1 + {\varOmega ^2}{t^2}} \right) }^{{{\left( {1 - s} \right) } / 2}}}\cos \left[ {\left( {s - 1} \right) \arctan \left( {\varOmega t} \right) } \right] } \right\} . \end{aligned}$$

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