Robustness measure of hybrid intra-particle entanglement, discord, and classical correlation with initial Werner state

Abstract

Quantum information processing is largely dependent on the robustness of non-classical correlations, such as entanglement and quantum discord. However, all the realistic quantum systems are thermodynamically open and lose their coherence with time through environmental interaction. The time evolution of quantum entanglement, discord, and the respective classical correlation for a single, spin-1/2 particle under spin and energy degrees of freedom, with an initial Werner state, has been investigated in the present study. The present intra-particle system is considered to be easier to produce than its inter-particle counterpart. Experimentally, this type of system may be realized in the well-known Penning trap. The most stable correlation was identified through maximization of a system-specific global objective function. Quantum discord was found to be the most stable, followed by the classical correlation. Moreover, all the correlations were observed to attain highest robustness under initial Bell state, with minimum possible dephasing and decoherence parameters.

Appendix: The quantum Liouville equation and its solution for a single spin-1/2 particle

$$\begin{aligned}&\mathop {\rho _{11} }\limits ^{\bullet } +\gamma \rho _{11} =0\nonumber \\&\quad \mathop {\rho _{12} }\limits ^{\bullet } +\left( {i\alpha \omega _0 +\gamma +\frac{\varGamma }{2}} \right) \rho _{12} =0\nonumber \\&\quad \mathop {\rho _{13} }\limits ^{\bullet } +\left( {i\omega _0 +\frac{\gamma }{2}} \right) \rho _{13} =0\nonumber \\&\quad \mathop {\rho _{14} }\limits ^{\bullet } +\left( {\left( {1+\alpha } \right) i\omega _0 +\frac{\gamma }{2}+\frac{\varGamma }{2}} \right) \rho _{14} =0\nonumber \\&\quad \mathop {\rho _{22} }\limits ^{\bullet } +\gamma \rho _{22} =0\nonumber \\&\quad \mathop {\rho _{23} }\limits ^{\bullet } +\left( {\left( {1-\alpha } \right) i\omega _0 +\frac{\gamma }{2}+\frac{\varGamma }{2}} \right) \rho _{23} =0\nonumber \\&\quad \mathop {\rho _{24} }\limits ^{\bullet } +\left( {i\omega _0 +\frac{\gamma }{2}} \right) \rho _{24} =0\nonumber \\&\quad \mathop {\rho _{33} }\limits ^{\bullet } -\gamma \rho _{11} =0\nonumber \\&\quad \mathop {\rho _{34} }\limits ^{\bullet } +\left( {i\alpha \omega _0 +\frac{\varGamma }{2}} \right) \rho _{34} -\gamma \rho _{12} =0\nonumber \\&\quad \mathop {\rho _{44} }\limits ^{\bullet } -\gamma \rho _{22} =0 \end{aligned}$$

(26)

Equation (26) was analytically solved in congruence with Eq. (2) as

$$\begin{aligned} \rho _{11} \left( t \right)= & {} \rho _{11} \left( 0 \right) e^{-\gamma t}\nonumber \\ \rho _{12} \left( t \right)= & {} \rho _{12} \left( 0 \right) e^{-\left( {\gamma +\frac{\varGamma }{2}+i\alpha \omega _0 } \right) t} \cdot \nonumber \\ \rho _{13} \left( t \right)= & {} \rho _{13} \left( 0 \right) e^{-\left( {\frac{\gamma }{2}+i\omega _0 } \right) t}\nonumber \\ \rho _{14} \left( t \right)= & {} \rho _{14} \left( 0 \right) e^{-\left( {\frac{\gamma }{2}+\frac{\varGamma }{2}+\left( {1+\alpha } \right) i\omega _0 } \right) t}\nonumber \\ \rho _{22} \left( t \right)= & {} \rho _{22} \left( 0 \right) e^{-\gamma t}\nonumber \\ \rho _{23} \left( t \right)= & {} \rho _{23} \left( 0 \right) e^{-\left( {\frac{\gamma }{2}+\frac{\varGamma }{2}+\left( {1-\alpha } \right) i\omega _0 } \right) t}\nonumber \\ \rho _{24} \left( t \right)= & {} \rho _{24} \left( 0 \right) e^{-\left( {\frac{\gamma }{2}+i\omega _0 } \right) t}\nonumber \\ \rho _{33} \left( t \right)= & {} \rho _{33} \left( 0 \right) +\left( {-e^{-\gamma t}+1} \right) \rho _{11} \left( 0 \right) \nonumber \\ \rho _{34} \left( t \right)= & {} \rho _{34} \left( 0 \right) e^{-\left( {\frac{\varGamma }{2}+i\alpha \omega _0 } \right) t}+\rho _{12} \left( 0 \right) e^{-\left( {\frac{\varGamma }{2}+i\alpha \omega _0 } \right) t}\left( {1-e^{-\gamma t}} \right) \nonumber \\ \rho _{44} \left( t \right)= & {} \rho _{44} \left( 0 \right) +\left( {-e^{-\gamma t}+1} \right) \rho _{22} \left( 0 \right) \end{aligned}$$

(27)

The lower triangular elements of the density operator can be obtained from Hermitian property$\left( {\rho _{ij} ^{*}=\rho _{ji} } \right)$.