# 热环境下超量子失协的动力学特性Dynamic Characteristics of Super Quantum Discord in Thermal Environment

DOI: 10.12677/MP.2020.106015, PDF, HTML, XML, 下载: 82  浏览: 170

Abstract: Based on the weak measurement theory, this paper gives the dynamics of quantum discord be-tween two isolated atoms in their respective thermal reservoirs, and analyzes the differences be-tween quantum discord and superquantum discord in the evolution process with time, as well as the factors affecting the dynamic evolution. We find that quantum correlation depends on the dis-turbance degree of quantum system, and the difference between standard quantum discord and super quantum discord caused by weak measurement increases with the decrease of measurement intensity parameter. This means that weak measurement can capture more quantum discord of two-qubit systems. Our results show that the weak measurement performed on one of the subsystems can lead to super quantum discord, which is a more natural quantum correlation measurement than the standard quantum discord captured by projection measurement.

1. 引言

2. 理论描述

2.1. 强测量下的量子失协

$QD\left({\rho }_{ab}\right)=I\left({\rho }_{ab}\right)-{C}^{\prime }\left({\rho }_{ab}\right)$ (1)

$I\left({\rho }_{ab}\right)=S\left({\rho }_{a}\right)+S\left({\rho }_{b}\right)-S\left({\rho }_{ab}\right)$ (2)

$S\left({\rho }_{j}\right)$ 是量子系统的Von Neumann熵，它描述了子系统a或b或者整个复合系统的不确定性：

$S\left({\rho }_{j}\right)=-t{r}_{j}\left({\rho }_{j}{\mathrm{log}}_{2}{\rho }_{j}\right)=-{\sum }_{i}{\lambda }_{j}^{i}{\mathrm{log}}_{2}{\lambda }_{j}^{i}$ (3)

${C}^{\prime }\left({\rho }_{ab}\right)=S\left({\rho }_{a}\right)-{\mathrm{min}}_{\left\{{\Pi }_{k}\right\}}\left[S\left({\rho }_{ab}|\left\{{\Pi }_{k}\right\}\right)\right]$ (4)

$QD\left({\rho }_{ab}\right)=S\left({\rho }_{b}\right)-S\left({\rho }_{ab}\right)+{\mathrm{min}}_{\left\{{\Pi }_{k}\right\}}S\left({\rho }_{ab}|\left\{{\Pi }_{k}\right\}\right)$ (5)

2.2. 弱测量下的超量子失协

$P\left(±x\right)=\sqrt{\frac{\left(1\mp \mathrm{tanh}x\right)}{2}}{\Pi }_{0}+\sqrt{\frac{\left(1±\mathrm{tanh}x\right)}{2}}{\Pi }_{1}$ (6)

${\rho }_{a|{P}^{b}\left(±x\right)}=\frac{t{r}_{b}\left[\left(I\otimes {P}^{b}\left(±x\right)\right){\rho }_{ab}\left(I\otimes {P}^{b}\left(±x\right)\right)\right]}{t{r}_{ab}\left[\left(I\otimes {P}^{b}\left(±x\right)\right){\rho }_{ab}\left(I\otimes {P}^{b}\left(±x\right)\right)\right]}$ (7)

$f\left(±x\right)=t{r}_{ab}\left[\left(I\otimes {P}^{b}\left(±x\right)\right){\rho }_{ab}\left(I\otimes {P}^{b}\left(±x\right)\right)\right]$ (8)

${S}_{w}\left({\rho }_{ab}|\left\{{P}^{b}\left(x\right)\right\}\right)=f\left(x\right)S\left({\rho }_{ab}|\left\{{P}^{b}\left(x\right)\right\}\right)+f\left(-x\right)S\left({\rho }_{ab}|\left\{{P}^{b}\left(-x\right)\right\}\right)$ (9)

$\text{SQD}\left({\rho }_{ab}\right)={\mathrm{min}}_{\left\{{\Pi }_{i}^{b}\right\}}{S}_{w}\left({\rho }_{ab}|\left\{{P}^{b}\left(x\right)\right\}\right)-S\left(a|b\right)$ (10)

$\text{SQD}\left({\rho }_{ab}\right)=S\left({\rho }_{b}\right)-S\left({\rho }_{ab}\right)+{\mathrm{min}}_{\left\{{\Pi }_{i}^{b}\right\}}{S}_{w}\left({\rho }_{ab}|\left\{{P}^{b}\left(x\right)\right\}\right)$ (11)

3. 分析与讨论

$\begin{array}{c}\stackrel{˙}{\rho }={\sum }_{i=1,2}\left\{-\frac{1}{2}{\gamma }_{i}\left({n}_{i}+1\right)\left[{\sigma }_{+}^{i}{\sigma }_{-}^{i}\rho -2{\sigma }_{-}^{i}\rho {\sigma }_{+}^{i}+\rho {\sigma }_{+}^{i}{\sigma }_{-}^{i}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{1}{2}{\gamma }_{i}{n}_{i}\left[{\sigma }_{-}^{i}{\sigma }_{+}^{i}\rho -2{\sigma }_{+}^{i}\rho {\sigma }_{-}^{i}+\rho {\sigma }_{-}^{i}{\sigma }_{+}^{i}\right]\right\}\end{array}$ (12)

${\rho }_{AB}\left(0\right)=\left(1-P\right)|11〉〈11|+P|{\phi }^{+}〉〈{\phi }^{+}|$ (13)

$\begin{array}{c}{\rho }_{11}\left(t\right)=\frac{1}{2{\left(2n+1\right)}^{2}}\left\{2{n}^{2}-2n\left[P\left(2n+1\right)-2\left(n+1\right)\right]{\text{e}}^{-\left(2n+1\right)\gamma t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left[P\left(n+1\right)\left(4n+2\right)-2{\left(n+1\right)}^{2}\right]{\text{e}}^{-2\left(2n+1\right)\gamma t}\right\}\end{array}$

$\begin{array}{l}{\rho }_{22}\left(t\right)={\rho }_{33}\left(t\right)=\frac{1}{2{\left(2n+1\right)}^{2}}\left\{2n\left(n+1\right)-\left[P\left(2n+1\right)-2\left(n+1\right)\right]{\text{e}}^{-\left(2n+1\right)\gamma t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\left[P\left(n+1\right)\left(4n+2\right)-2{\left(n+1\right)}^{2}\right]{\text{e}}^{-2\left(2n+1\right)\gamma t}\right\}\end{array}$

$\begin{array}{c}{\rho }_{44}\left(t\right)=\frac{1}{2{\left(2n+1\right)}^{2}}\left\{2{\left(n+1\right)}^{2}+2\left(n+1\right)\left[P\left(2n+1\right)-2\left(n+1\right)\right]{\text{e}}^{-\left(2n+1\right)\gamma t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left[P\left(n+1\right)\left(4n+2\right)-2{\left(n+1\right)}^{2}\right]{\text{e}}^{-2\left(2n+1\right)\gamma t}\right\}\end{array}$

${\rho }_{23}\left(t\right)={\rho }_{32}\left(t\right)=\frac{P}{2}{\text{e}}^{-\left(2n+1\right)\gamma t}$ (14)

(a) (b) (c)

Figure 1. Dynamic evolution of two-atom quantum discord and superquantum discord. Solid line (red) represents super quantum discord, and dotted line (blue) represents quantum discord. In which parameters P = 0.5, x = 1 (a) n = 0.01; (b) n = 1; (c) n = 5

(a) (b) (c)

Figure 2. Dynamic evolution of two-atom quantum discord and superquantum discord. Solid line (red) represents super quantum discord, and dotted line (blue) represents quantum discord. In which parameters n = 0.1, x = 1 (a) P = 0.1; (b) P = 0.3; (c) P = 0.99

(a) (b) (c)

Figure 3. Dynamic evolution of two-atom quantum discord and superquantum discord. Solid line (red) represents super quantum discord, and dotted line (blue) represents quantum discord. In which parameters P = 0.5, n = 0.1 (a ) x = 0.1; (b ) x = 1; (c ) x = 5

4. 结论

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