Entropic uncertainty relation in a two-qutrit system with external magnetic field and Dzyaloshinskii–Moriya interaction under intrinsic decoherence

Abstract

In this paper, we explore the dynamic behaviors of entropic uncertainty relation in a two-qutrit system which is in the presence of external magnetic field and Dzyaloshinskii–Moriya (DM) interaction under intrinsic decoherence. The effects of the isotropic bilinear interaction, the external magnetic field, the DM interaction strength, as well as the intrinsic decoherence on the entropic uncertainty relation have been demonstrated in detail. Compared with previous results, our results show that, controlling the isotropic bilinear interaction parameter J, the external magnetic field strength $B_{0}$, the DM interaction parameter D can result in inflation of the uncertainty, while increasing the intrinsic decoherence parameter can lift the uncertainty of the measurement. In particularly, under certain conditions (e.g., parameters J, $B_{0}$ and D are large enough), the entropic uncertainty will ultimately tend to a stable value and be immune to decoherence.

Appendix A

In this Appendix, we give the explicit analytic forms of the entropic uncertainty Eq.(10) and its lower bound Eq.(11) in the text. If one chooses two of the spin-1 observables $S_{x}$ and $S_{z}$ whose eigenbases correspond to $(1/2,\sqrt{2}/2,1/2)^\mathrm{{T}}$, $(1/2,-\sqrt{2}/2,1/2)^\mathrm{{T}}$ and $(-\sqrt{2}/2,0,\sqrt{2}/2)^\mathrm{{T}}$ , as well as $(1,0,0)^\mathrm{{T}}$, $(0,1,0)^\mathrm{{T}}$ and $(0,0,1)^\mathrm{{T}}$, respectively, to perform on qutrit A. The post-measurement states $\rho _{X|B}=\sum _{X}(|\phi _{X}\rangle \langle \phi _{X}|\otimes I)\rho _{AB}(|\phi _{X}\rangle \langle \phi _{X}|\otimes I)$ have the forms, respectively,

$$\begin{aligned} \rho (S_{x}|B)= & {} \left( \begin{array}{ c c c c c c c c c c l r } \frac{19+2M}{168} &{} 0 &{} 0 &{} 0 &{} \frac{\eta \nu }{84} &{} \frac{1}{84\Delta }N^{*} &{} \frac{2M-1}{168} &{} 0 &{} 0\\ \\ 0 &{} \frac{16+\alpha -3M}{168} &{} -\frac{1}{168\Delta }E^{*} &{} \frac{\eta ^{*} \nu ^{*}}{84}&{} 0 &{} 0 &{} 0 &{} \frac{8-3\alpha +M}{168} &{} +\frac{1}{56\Delta }E^{*}\\ \\ 0 &{}-\frac{1}{168\Delta }E &{} \frac{21-\alpha }{168} &{} \frac{1}{84\Delta }N &{} 0 &{} 0 &{} 0 &{} \frac{1}{56\Delta }E &{} \frac{3\alpha -7}{168}\\ \\ 0 &{} \frac{\eta \nu }{84} &{} \frac{1}{84\Delta }N^{*} &{} \frac{9+2M}{84} &{} 0 &{} 0 &{} 0 &{} \frac{\eta \nu }{84} &{} \frac{1}{84\Delta }N^{*} \\ \\ \frac{\eta ^{*} \nu ^{*}}{84} &{} 0 &{} 0 &{} 0 &{} \frac{12-\alpha -M}{84} &{} \frac{1}{84\Delta }E^{*} &{} \frac{\eta ^{*} \nu ^{*}}{84} &{} 0 &{} 0\\ \\ \frac{1}{84\Delta }N &{} 0 &{} 0 &{} 0 &{} \frac{1}{84\Delta }E &{} \frac{7+\alpha }{84} &{} \frac{1}{84\Delta }N &{} 0 &{} 0\\ \\ \frac{2M-1}{168} &{} 0 &{} 0&{} 0 &{} \frac{\eta \nu }{84} &{} \frac{1}{84\Delta }N^{*} &{} \frac{19+2M}{168} &{} 0 &{} 0\\ \\ 0 &{} \frac{8-3\alpha +M}{168} &{} \frac{1}{56\Delta }E^{*} &{} \frac{\eta ^{*} \nu ^{*}}{84} &{} 0 &{} 0 &{} 0 &{} \frac{16+\alpha -3M}{168} &{} -\frac{1}{168\Delta }E^{*}\\ \\ 0 &{} \frac{1}{56\Delta }E &{} \frac{3\alpha -7}{168} &{} \frac{1}{84\Delta }N &{} 0 &{} 0 &{} 0 &{} -\frac{1}{168\Delta }E &{} \frac{21-\alpha }{168}\\ \\ \end{array} \right) ,\nonumber \\ \end{aligned}$$

(A1)

$$\begin{aligned} \rho (S_{z}|B)= & {} \frac{\alpha }{21}(|00\rangle \langle 00|+|12\rangle \langle 12|+|21\rangle \langle 21|)\nonumber \\&+\,\frac{5-\alpha }{21}(|02\rangle \langle 02|+|11\rangle \langle 11|+|20\rangle \langle 20|)\nonumber \\&+\,\frac{1}{21}(2-M)|01\rangle \langle 01|+\frac{1}{21}(2+M)|10\rangle \langle 10|+\frac{2}{21}|22\rangle \langle 22| \end{aligned}$$

(A2)

where $M=\eta \epsilon +\eta ^{*}\epsilon ^{*}$,

$$\begin{aligned} N= & {} \varepsilon _{-} \chi _{-}+\varepsilon _{+} \mu _{+},\\ E= & {} \varepsilon _{-} \mu _{-}+\varepsilon _{+} \chi _{+}. \end{aligned}$$

According to Eq.(2), the left hand and the right hand of the entropic uncertainty relation reduce to

$$\begin{aligned} U= & {} \frac{1}{21}\left[ \log _{2}{\frac{(49-M^2)^{14}}{20502297788(5-\alpha ) ^{15}(4-M^2)^{2}}}+6\alpha Arc\tanh \left( 1-\frac{2\alpha }{5}\right) \right. \nonumber \\&\left. +\,4M Arc \tanh \left( \frac{M}{7}\right) -2M Arc\tanh \left( \frac{M}{2}\right) \right] \nonumber \\&-\,\sum _{i}\omega _{i}\log _{2}\omega _{i} \end{aligned}$$

(A3)

$$\begin{aligned} U_{b}= & {} 1+\frac{1}{21}\left[ (7-M)\log _{2}{(7-M)}+(7+M)\log _{2}{(7+M)} -7\log _{2}{1323}\right] \nonumber \\&-\,\sum _{i}\lambda _{i}\log _{2}\lambda _{i}. \end{aligned}$$

(A4)