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Analytic expressions of quantum correlations in qutrit Werner states

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Abstract

Quantum correlations in qutrit Werner states are extensively investigated with five popular methods, namely, original quantum discord (OQD) (Ollivier and Zurek in Phys Rev Lett 88:017901, 2001), measurement-induced disturbance (MID) (Luo in Phys Rev A 77:022301, 2008), ameliorated MID (AMID) (Girolami et al. in J Phys A Math Theor 44:352002, 2011), relative entropy (RE) (Modi et al. in Phys Rev Lett 104:080501, 2010) and geometric discord (GD) (Dakić et al. in Phys Rev Lett 105:190502, 2010). Two different analytic expressions of quantum correlations are derived. Quantum correlations captured by the former four methods are same and bigger than those obtained via the GD method. Nonetheless, they all qualitatively characterize quantum correlations in the concerned states. Moreover, as same as the qubit case, there exist quantum correlations in separable qutrit Werner states, too.

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Acknowledgments

Supported by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20103401110007, the NSFC under Grant Nos. 10874122, 10975001, 51072002 and 51272003, the Program for Excellent Talents at the University of Guangdong province (Guangdong Teacher Letter [1010] No. 79), and the 211 Project of Anhui University.

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Authors and Affiliations

  1. School of Physics and Material Science, Anhui University, Hefei, 230039, Anhui, China

    Biaoliang Ye, Jianlan Chen, Xiansong Liu & Zhanjun Zhang

  2. Department of Physics, Shaoguan University, Shaoguan, 512005, China

    Yimin Liu

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  1. Biaoliang Ye
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  2. Yimin Liu
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Correspondence to Zhanjun Zhang.

Appendix

Appendix

Some partial derivatives

$$\begin{aligned} \mathcal{R}(u_{ij}) : = \log _2 \left( 1-z + 3z|u_{ij}|^2\right)/9+\frac{1}{\ln 2}, \quad i,j=0,1,2. \end{aligned}$$
(38)
$$\begin{aligned}&\frac{\partial S\left(\rho _{AB}|\{\Pi _{A}^{(k)}\}\right)}{\partial {\theta _1}}\nonumber \\&\quad = \frac{2z}{3}\left\{ -\sin 2\theta _1\cos ^2\theta _2\mathcal{R}(u_{00}) +\sin 2\theta _1\mathcal{R}(u_{01}) -\sin 2\theta _1\sin ^2\theta _2\mathcal{R}(u_{02})\right. \nonumber \\&\quad \quad + \left(\sin 2\theta _1\cos ^2\theta _2\cos ^2\theta _3 -\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4\right)\mathcal{R}(u_{10})\nonumber \\&\quad \quad -\sin 2\theta _1\cos ^2\theta _3\mathcal{R}(u_{11}) \nonumber \\&\quad \quad + \left(\sin 2\theta _1\sin ^2\theta _2\cos ^2\theta _3+\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4\right)\mathcal{R}(u_{12})\nonumber \\&\quad \quad +\left(\sin 2\theta _1\cos ^2\theta _2\sin ^2\theta _3+\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4\right)\mathcal{R}(u_{20})\nonumber \\&\quad \quad -\sin 2\theta _1\sin ^2\theta _3\mathcal{R}(u_{21})\nonumber \\&\quad \quad \left.+\left(\sin 2\theta _1\sin ^2\theta _2\sin ^2\theta _3-\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4\right)\mathcal{R}(u_{22})\right\} , \end{aligned}$$
(39)
$$\begin{aligned}&\frac{\partial S\left(\rho _{AB}|\big \{\Pi _{A}^{(k)}\big \}\right)}{\partial {\theta _2}}\nonumber \\&= \frac{2z}{3}\left\{ -\cos ^2\theta _1\sin 2\theta _2\mathcal{R}(u_{00})+\cos ^2\theta _1\sin 2\theta _2\mathcal{R}(u_{02})\right.\nonumber \\&\quad \quad +\left[\sin 2\theta _2 \big (\sin ^2\theta _3-\sin ^2\theta _1\cos ^2\theta _3\big )-\sin \theta _1\cos 2\theta _2\sin 2\theta _3\cos \phi _4\right] (\mathcal{R}(u_{10})-\mathcal{R}(u_{12}))\nonumber \\&\quad \quad \left.+\left[\sin 2\theta _2\big (\cos ^2\theta _3-\sin ^2\theta _1\sin ^2\theta _3\big ) +\sin \theta _1\cos 2\theta _2\sin 2\theta _3\cos \phi _4\right]\big (\mathcal{R}(u_{20})-\mathcal{R}(u_{22})\big )\right\} ,\nonumber \\ \end{aligned}$$
(40)
$$\begin{aligned}&\frac{\partial S\left(\rho _{AB}|\big \{\Pi _{A}^{(k)}\big \}\right)}{\partial {\theta _3}}\nonumber \\&\quad = \frac{2z}{3} \left\{ \left[(\sin ^2\theta _2-\sin ^2\theta _1\cos ^2\theta _2)\sin 2\theta _3-\sin \theta _1\sin 2\theta _2\cos 2\theta _3\cos \phi _4\right]\mathcal{R}(u_{10})\right.\nonumber \\&\quad \quad -\cos ^2\theta _1\sin 2\theta _3\mathcal{R}(u_{11}) +\left[\big (\cos ^2\theta _2-\sin ^2\theta _1\sin ^2\theta _2\big ) \sin 2\theta _3\right.\nonumber \\&\quad \quad +\left.\sin \theta _1\sin 2\theta _2\cos 2\theta _3\cos \phi _4\right] \mathcal{R}(u_{12}) \nonumber \\&\quad \quad + \left[\left(\sin ^2\theta _1\cos ^2\theta _2 -\sin ^2\theta _2\right) \sin 2\theta _3+\sin \theta _1\sin 2\theta _2\cos 2\theta _3\cos \phi _4\right] \mathcal{R}(u_{20})\nonumber \\&\quad \quad +\cos ^2\theta _1\sin 2\theta _3\mathcal{R}(u_{21})\nonumber \\&\quad \quad \left.+ \left[\big (\sin ^2\theta _1\sin ^2\theta _2-\cos ^2\theta _2\big ) \sin 2\theta _3-\sin \theta _1\sin 2\theta _2\cos 2\theta _3\cos \phi _4\right]\mathcal{R}(u_{22})\right\} ,\nonumber \\ \end{aligned}$$
(41)
$$\begin{aligned}&\frac{\partial S\left(\rho _{AB}|\big \{\Pi _{A}^{(k)}\big \}\right)}{\partial {\phi _4}}\nonumber \\&\quad \!= \!\frac{2z}{3}\left\{ \frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4\mathcal{R}(u_{10}) \!-\!\frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4\mathcal{R}(u_{12})\right.\nonumber \\&\quad \quad \left.-\frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4\mathcal{R}(u_{20}) \!+\!\frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4\mathcal{R}(u_{22})\right\} .\nonumber \\ \end{aligned}$$
(42)

If \(\theta _1=\theta _2=\theta _3=\phi _4=0\), then \(\frac{\partial S(\rho _{AB}|\{\Pi _{A}^j\})}{\partial \theta _1} =\frac{\partial S(\rho _{AB}|\{\Pi _{A}^j\})}{\partial \theta _2}=\frac{\partial S(\rho _{AB}|\{\Pi _{A}^j\})}{\partial \theta _3} =\frac{\partial S(\rho _{AB}|\{\Pi _{A}^j\})}{\partial \phi _4}=0\).

$$\begin{aligned}&\frac{\partial \mathcal{Q}_G(\rho _{AB})}{\partial {\theta _1}}\nonumber \\&\quad =-\frac{z^2}{9}\left[-\sin 2\theta _1\big (\cos ^2\theta _2+1\big ) \big (|u_{00}|^2-|u_{01}|^2\big ) +\left[\sin 2\theta _1\big ( \cos ^2\theta _2+1\big )\cos ^2\theta _3\right.\right.\nonumber \\&\quad \quad \left.-\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4\right] \big (|u_{10}|^2-|u_{11}|^2\big ) +\left[\sin 2\theta _1\big (\cos ^2\theta _2+1\big )\sin ^2\theta _3\right.\nonumber \\&\quad \quad \left.+\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4\right] \big (|u_{20}|^2-|u_{21}|^2\big )+\sin 2\theta _1\cos ^2\theta _2\big (1-3|u_{02}|^2\big )\nonumber \\&\quad \quad -\big (\sin 2\theta _1\sin ^2\theta _2\cos ^2\theta _3+ \frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4\big )\big (1-3|u_{12}|^2\big )\nonumber \\&\quad \quad \left.-\big (\sin 2\theta _1\sin ^2\theta _2\sin ^2\theta _3-\frac{1}{2}\cos \theta _1\sin 2\theta _2\sin 2\theta _3\cos \phi _4\big )\big (1-3|u_{22}|^2\big )\right], \end{aligned}$$
(43)
$$\begin{aligned}&\frac{\partial \mathcal{Q}_G(\rho _{AB})}{\partial {\theta _2}}\nonumber \\&=-\frac{z^2}{9} \left\{ \big (-\cos ^2\theta _1\sin 2\theta _2\big )\big (|u_{00}|^2-|u_{01}|^2\big ) +\left[\sin 2\theta _2\big (\sin ^2\theta _3-\sin ^2\theta _1\cos ^2\theta _3\big )\right.\right.\nonumber \\&\quad \left.-\sin \theta _1\cos 2\theta _2\sin 2\theta _3\cos \phi _4\right]\big (|u_{10}|^2-|u_{11}|^2\big )+ \left[\sin 2\theta _2(\cos ^2\theta _3-\sin ^2\theta _1\sin ^2\theta _3)\right.\nonumber \\&\quad \left.+\sin \theta _1\cos 2\theta _2\sin 2\theta _3\cos \phi _4\right] \big (|u_{20}|^2-|u_{21}|^2\big )-\cos ^2\theta _1\sin 2\theta _2(1-3|u_{02}|^2)\nonumber \\&\quad -\left[\sin 2\theta _2 \big (\sin ^2\theta _1\cos ^2\theta _3-\sin ^2\theta _3\big )+\sin \theta _1\cos 2\theta _2\sin 2\theta _3\cos \phi _4\right]\big (1-3|u_{12}|^2\big )\nonumber \\&\quad \left.-\left[\sin 2\theta _2\big (\sin ^2\theta _1\sin ^2\theta _3-\cos ^2\theta _3\big )- \sin \theta _1\cos 2\theta _2\sin 2\theta _3\cos \phi _4\right]\big (1-3|u_{22}|^2\big )\right\} , \end{aligned}$$
(44)
$$\begin{aligned}&\frac{\partial \mathcal{Q}_G(\rho _{AB})}{\partial {\theta _3}}\nonumber \\&=-\frac{z^2}{9} \left\{ \left[ \big (\cos ^2\theta _1+\sin ^2\theta _2\right.\right.\nonumber \\&\quad \quad -\left.\sin ^2\theta _1\cos ^2\theta _2\big ) \sin 2\theta _3-\sin \theta _1\sin 2\theta _2\cos 2\theta _3\cos \phi _4\right] \nonumber \\&\quad \quad \left[|u_{10}|^2+|u_{21}|^2-|u_{11}|^2-|u_{20}|^2\right]\nonumber \\&\quad \quad +3\left[\big (\cos ^2\theta _2-\sin ^2\theta _1\sin ^2\theta _2\big )\sin 2\theta _3\right.\nonumber \\&\quad \quad \left.\left.+\sin \theta _1\sin 2\theta _2\cos 2\theta _3\cos \phi _4\right] \big (|u_{12}|^2-|u_{22}|^2\big )\right\} , \end{aligned}$$
(45)
$$\begin{aligned}&\frac{\partial \mathcal{Q}_G(\rho _{AB})}{\partial {\phi _4}}\nonumber \\&=-\frac{z^2}{9}\left[\frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4\big (|u_{10}|^2-|u_{11}|^2\big )\right.\nonumber \\&\quad \quad -\frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4\big (|u_{20}|^2-|u_{21}|^2\big )\nonumber \\&\quad \quad +\frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4\big (1-3|u_{12}|^2\big )\nonumber \\&\quad \quad \left.- \frac{1}{2}\sin \theta _1\sin 2\theta _2\sin 2\theta _3\sin \phi _4\big (1-3|u_{22}|^2\big )\right]. \end{aligned}$$
(46)

It is easy to see that \(\frac{\partial \mathcal{Q}_G(\rho _{AB})}{\partial {\theta _1}}= \frac{\partial \mathcal{Q}_G(\rho _{AB})}{\partial {\theta _2}}= \frac{\partial \mathcal{Q}_G(\rho _{AB})}{\partial {\theta _3}}= \frac{\partial \mathcal{Q}_G(\rho _{AB})}{\partial {\phi _4}}=0\) hold if \(\theta _1=\theta _2=\theta _3=\phi _4=0\).

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Ye, B., Liu, Y., Chen, J. et al. Analytic expressions of quantum correlations in qutrit Werner states. Quantum Inf Process 12, 2355–2369 (2013). https://doi.org/10.1007/s11128-013-0531-y

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