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Comparative study of the dynamics of entanglement and purity in two hybrid quantum systems under the effect of static noise

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Abstract

A comparative study of the dynamics of entanglement and purity in two hybrid quantum systems under the effect of a static noise is presented. The first system contains two qubits and one qutrit, while the second system contains one qubit and two qutrits. In addition, each system presents three configurations: the different, the common and the bipartite environment. In the bipartite environment, we have three cases depending on whether two subsystems are in the same environment and one subsystem in its environment alone. We found that in all these configurations, the same behavior is observed for negativity, purities and degrees of purity. The lower the purity, the more the degree of purity becomes and the system is less influenced by environmental corruption. Purity decreases more in the second system than in the first system and we can conclude that the second system preserves quantum properties better than the first.

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Authors and Affiliations

  1. Unité de Recherche de Matière Condensée, d´Electronique Et de Traitement du Signal (URMACETS), Department of Physics, Dschang School of Sciences and Technology, University of Dschang, P.O. Box 67, Dschang, Cameroon

    Lionel Tenemeza Kenfack, Martin Tchoffo & Lukong Cornelius Fai

  2. Department of Physics, Faculty of Science, University of Maroua, P.O. Box 814, Maroua, Cameroon

    Odette Melachio Tiokang, Fridolin Nya Tchangnwa & Jaures Diffo Tchinda

  3. Faculté D’agronomie Et Des Sciences Agricoles, Centre d’Etudes Et de Recherches en Agronomie Et en Biodiversité (CERAB), University of Dschang, P.O. 222, Dschang, Cameroon

    Martin Tchoffo

Authors
  1. Odette Melachio Tiokang
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  2. Fridolin Nya Tchangnwa
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  3. Jaures Diffo Tchinda
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  4. Lionel Tenemeza Kenfack
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  5. Martin Tchoffo
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  6. Lukong Cornelius Fai
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Correspondence to Martin Tchoffo.

Appendices

Appendix A Explicit derivation of the density matrix of the two systems in the different environment, common and bipartite environments

  1. A

    First system: two qubits and one qutrit

  1. (A1)

    Independent environments (de)

For the case of local coupling to different environments the density matrix \(\rho_{1de} (t),\) of the system at time t, reads:

$$ \rho_{1de} (t) = \left( {\begin{array}{*{20}c} {\rho_{11d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{121d}^{ + } } \\ 0 & {\rho_{11d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{112d}^{ + } } & 0 \\ 0 & 0 & {\rho_{33d} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{103d}^{ + } } & 0 & 0 \\ 0 & 0 & 0 & {\rho_{11d} } & 0 & 0 & 0 & 0 & {\rho_{94d}^{ + } } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{11d} } & 0 & 0 & {\rho_{85d}^{ + } } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{33d} } & {\rho_{76d}^{ + } } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{76d} } & {\rho_{33d} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{85d} } & 0 & 0 & {\rho_{11d} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho_{94d} } & 0 & 0 & 0 & 0 & {\rho_{11d} } & 0 & 0 & 0 \\ 0 & 0 & {\rho_{103d} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33d} } & 0 & 0 \\ 0 & {\rho_{112d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11d} } & 0 \\ {\rho_{121d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11d} } \\ \end{array} } \right) $$
$$ \rho_{11d} = \frac{p}{4};k3\rho_{33d} = \frac{1}{4}\left( {1 - 2p} \right); $$
$$ \begin{aligned} \rho_{121d} = & \frac{p}{{4\left( {\Delta_{c} t} \right)^{3} J_{1} J_{2} J_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} + c_{0} J_{1} + c_{0} J_{2} + c_{0} J_{3} } \right)t} \right\} \\ & \quad \sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right)\sin \left( {\Delta_{c} tJ_{3} } \right) \\ \end{aligned} $$
$$ \rho_{112d} = \frac{p}{{4\left( {\Delta_{c} t} \right)^{2} J_{1} J_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + c_{0} J_{1} + c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
$$ \begin{aligned} \rho_{103d} =\, & \frac{{\left( {1 - 2p} \right)}}{{4\left( {\Delta_{c} t} \right)^{3} J_{1} J_{2} J_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} - \varepsilon_{3} + c_{0} J_{1} + c_{0} J_{2} - c_{0} J_{3} } \right)t} \right\} \\ & \quad \sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right)\sin \left( {\Delta_{c} tJ_{3} } \right) \\ \end{aligned} $$
$$ \begin{aligned} \rho_{94d} =\, & \frac{p}{{4\left( {\Delta_{c} t} \right)^{3} J_{1} J_{2} J_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} + c_{0} J_{1} - c_{0} J_{2} + c_{0} J_{3} } \right)t} \right\} \\ & \quad \sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right)\sin \left( {\Delta_{c} tJ_{3} } \right) \\ \end{aligned} $$
$$ \rho_{85d} =\, \frac{p}{{4\left( {\Delta_{c} t} \right)^{2} J_{1} J_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + c_{0} J_{1} - c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
$$ \begin{aligned} \rho_{76d} =\, & \frac{{\left( {1 - 2p} \right)}}{{4\left( {\Delta_{c} t} \right)^{3} J_{1} J_{2} J_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} + c_{0} J_{1} - c_{0} J_{2} - c_{0} J_{3} } \right)t} \right\} \\ & \quad \sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right)\sin \left( {\Delta_{c} tJ_{3} } \right) \\ \end{aligned} $$
  1. (A2)

    Common environment (Ce)

$$ \rho_{1ce} (t) = \left( {\begin{array}{*{20}c} {\rho_{11c} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{121c}^{ + } } \\ 0 & {\rho_{11c} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{112c}^{ + } } & 0 \\ 0 & 0 & {\rho_{33c} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{103c}^{ + } } & 0 & 0 \\ 0 & 0 & 0 & {\rho_{11c} } & 0 & 0 & 0 & 0 & {\rho_{94c}^{ + } } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{11c} } & 0 & 0 & {\rho_{85c}^{ + } } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{33c} } & {\rho_{76c}^{ + } } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{76c} } & {\rho_{33c} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{85c} } & 0 & 0 & {\rho_{11c} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho_{94c} } & 0 & 0 & 0 & 0 & {\rho_{11c} } & 0 & 0 & 0 \\ 0 & 0 & {\rho_{103c} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33c} } & 0 & 0 \\ 0 & {\rho_{112c} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11c} } & 0 \\ {\rho_{121c} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11c} } \\ \end{array} } \right) $$

With:

$$ \rho_{11c} = \frac{p}{4};\quad \rho_{33c} = \frac{1}{4}\left( {1 - 2p} \right) $$
$$ \rho_{121c} = \frac{p}{{\left( {12J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} + 3Jc_{0} } \right)t} \right\}\sin \left( {3Jt\Delta_{c} } \right) $$
$$ \rho_{112c} = \frac{p}{{\left( {8J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + 2Jc_{0} } \right)t} \right\}\sin \left( {2Jt\Delta_{c} } \right) $$
$$ \rho_{103c} = \frac{{\left( {1 - 2p} \right)}}{{\left( {4J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} - \varepsilon_{3} + Jc_{0} } \right)t} \right\}\sin \left( {Jt\Delta_{c} } \right) $$
$$ \rho_{94c} = \frac{p}{{\left( {4J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} + Jc_{0} } \right)t} \right\}\sin \left( {Jt\Delta_{c} } \right) $$
$$ \rho_{85c} = \frac{p}{4}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} } \right)t} \right\} $$
$$ \rho_{76c} = \frac{{\left( {1 - 2p} \right)}}{{\left( {4J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} - Jc_{0} } \right)t} \right\}\sin \left( {Jt\Delta_{c} } \right) $$
  1. (A3)

    Bipartite environments coupling (be)

In the case of bipartite environments, two subsystems interact with a common environment. We have envisaged three possibilities:

  1. a—

    The case where \(\left( {Q_{1} } \right)\) and \(\left( {Q_{2} } \right)\) are in common environment.

$$ \rho_{1Bea} (t) = \left( {\begin{array}{*{20}c} {\rho_{11Ba} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{121Ba}^{ + } } \\ 0 & {\rho_{11Ba} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{112Ba}^{ + } } & 0 \\ 0 & 0 & {\rho_{33Ba} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{103Ba}^{ + } } & 0 & 0 \\ 0 & 0 & 0 & {\rho_{11Ba} } & 0 & 0 & 0 & 0 & {\rho_{94Ba}^{ + } } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{11Ba} } & 0 & 0 & {\rho_{85Ba}^{ + } } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{33Ba} } & {\rho_{76Ba}^{ + } } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{76Ba} } & {\rho_{33Ba} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{85Ba} } & 0 & 0 & {\rho_{11Ba} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho_{94Ba} } & 0 & 0 & 0 & 0 & {\rho_{11Ba} } & 0 & 0 & 0 \\ 0 & 0 & {\rho_{103Ba} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33Ba} } & 0 & 0 \\ 0 & {\rho_{112Ba} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11Ba} } & 0 \\ {\rho_{121Ba} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11Ba} } \\ \end{array} } \right) $$

With:

$$ \rho_{11Ba} = \frac{p}{4};\quad \rho_{33Ba} = \frac{1}{4}\left( {1 - 2p} \right) $$
$$ \rho_{121Ba} = \frac{p}{{8\left( {\Delta_{c} t} \right)^{2} JJ_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} + 2c_{0} J + c_{0} J_{3} } \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{112Ba} = \frac{p}{{\left( {8\Delta_{c} Jt} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + 2c_{0} J} \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right) $$
$$ \rho_{103Ba} = \frac{{\left( {1 - 2p} \right)}}{{8\left( {\Delta_{c} t} \right)^{2} JJ_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} - \varepsilon_{3} + 2c_{0} J - c_{0} J_{3} } \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{94Ba} = \frac{p}{{\left( {4\Delta_{c} tJ_{3} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} + c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{85Ba} = \frac{p}{4}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} } \right)t} \right\} $$
$$ \rho_{76Ba} = \frac{{\left( {1 - 2p} \right)}}{{\left( {4\Delta_{c} tJ_{3} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} - c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{3} } \right) $$
  1. b—

    The case where \(\left( {Q_{1} } \right)\) and \(\left( {Q_{3} } \right)\) are in common environment.

$$ \rho_{1Beb} (t) = \left( {\begin{array}{*{20}c} {\rho_{11Bb} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{121Bb}^{ + } } \\ 0 & {\rho_{11Bb} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{112Bb}^{ + } } & 0 \\ 0 & 0 & {\rho_{33Bb} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{103Bb}^{ + } } & 0 & 0 \\ 0 & 0 & 0 & {\rho_{11Bb} } & 0 & 0 & 0 & 0 & {\rho_{94Bb}^{ + } } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{11Bb} } & 0 & 0 & {\rho_{85Bb}^{ + } } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{33Bb} } & {\rho_{76Bb}^{ + } } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{76Bb} } & {\rho_{33Bb} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{85Bb} } & 0 & 0 & {\rho_{11Bb} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho_{94Bb} } & 0 & 0 & 0 & 0 & {\rho_{11Bb} } & 0 & 0 & 0 \\ 0 & 0 & {\rho_{103Bb} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33Bb} } & 0 & 0 \\ 0 & {\rho_{112Bb} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11Bb} } & 0 \\ {\rho_{121Bb} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11Bb} } \\ \end{array} } \right) $$

With:

$$ \rho_{11Bb} = \frac{p}{4};\quad \rho_{33Bb} = \frac{1}{4}\left( {1 - 2p} \right) $$
$$ \rho_{121Bb} = \frac{p}{{8\left( {\Delta_{c} t} \right)^{2} JJ_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} + 2c_{0} J + c_{0} J_{2} } \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
$$ \rho_{112Bb} = \frac{p}{{4\left( {\Delta_{c} t} \right)^{2} JJ_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + c_{0} J + c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
$$ \rho_{103Bb} = \frac{{\left( {1 - 2p} \right)}}{{\left( {4\Delta_{c} tJ_{2} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} - \varepsilon_{3} + c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{2} } \right) $$
$$ \rho_{94Bb} = \frac{p}{{8\left( {\Delta_{c} t} \right)^{2} JJ_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} + 2c_{0} J - c_{0} J_{2} } \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
$$ \rho_{85Bb} = \frac{p}{{4\left( {\Delta_{c} t} \right)^{2} JJ_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + c_{0} J - c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
$$ \rho_{76Bb} = \frac{{\left( {1 - 2p} \right)}}{{\left( {4\Delta_{c} tJ_{2} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} - c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{2} } \right) $$
  1. c—

    The case where \(\left( {Q_{2} } \right)\) and \(\left( {Q_{3} } \right)\) are in common environment.

$$ \rho_{1Bec} (t) = \left( {\begin{array}{*{20}c} {\rho_{11Bc} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{121Bc}^{ + } } \\ 0 & {\rho_{11Bc} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{112Bc}^{ + } } & 0 \\ 0 & 0 & {\rho_{33Bc} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{103Bc}^{ + } } & 0 & 0 \\ 0 & 0 & 0 & {\rho_{11Bc} } & 0 & 0 & 0 & 0 & {\rho_{94Bc}^{ + } } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{11Bc} } & 0 & 0 & {\rho_{85Bc}^{ + } } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{33Bc} } & {\rho_{76Bc}^{ + } } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{76Bc} } & {\rho_{33Bc} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{85Bc} } & 0 & 0 & {\rho_{11Bc} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho_{94Bc} } & 0 & 0 & 0 & 0 & {\rho_{11Bc} } & 0 & 0 & 0 \\ 0 & 0 & {\rho_{103Bc} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33Bc} } & 0 & 0 \\ 0 & {\rho_{112Bc} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11Bc} } & 0 \\ {\rho_{121Bc} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11Bc} } \\ \end{array} } \right) $$

With:

$$ \rho_{11Bc} = \frac{p}{4};\quad \rho_{33Bc} = \frac{1}{4}\left( {1 - 2p} \right) $$
$$ \rho_{121Bc} = \frac{p}{{8\left( {\Delta_{c} t} \right)^{2} JJ_{1} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} + c_{0} J_{1} + 2c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {2\Delta_{c} tJ} \right) $$
$$ \rho_{112Bc} = \frac{p}{{4\left( {\Delta_{c} t} \right)^{2} J_{1} J}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + c_{0} J_{1} + c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ} \right) $$
$$ \rho_{103Bc} = \frac{{\left( {1 - 2p} \right)}}{{\left( {4\Delta_{c} tJ_{1} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} - \varepsilon_{3} + c_{0} J_{1} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right) $$
$$ \rho_{94Bc} = \frac{p}{{\left( {4\Delta_{c} tJ_{1} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} + c_{0} J_{1} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right) $$
$$ \rho_{85Bc} = \frac{p}{{4\left( {\Delta_{c} t} \right)^{2} J_{1} J}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + c_{0} J_{1} - c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ} \right) $$
$$ \rho_{76Bc} = \frac{{\left( {1 - 2p} \right)}}{{8\left( {\Delta_{c} t} \right)^{2} J_{1} J}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} + c_{0} J_{1} - 2c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {2\Delta_{c} tJ} \right) $$
  1. B—

    Second system: one qubit and two qutrits

  1. (B1)

    independent environments (de)

$$ \rho_{2de} (t) = \left( {\begin{array}{*{20}c} {\rho_{11d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{181d}^{ + } } \\ 0 & {\rho_{11d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{172d}^{ + } } & 0 \\ 0 & 0 & {\rho_{33d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{163d}^{ + } } & 0 & 0 \\ 0 & 0 & 0 & {\rho_{11d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{154d}^{ + } } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{11d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{145d}^{ + } } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{33d} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{136d}^{ + } } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11d} } & 0 & 0 & 0 & 0 & {\rho_{127d}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11d} } & 0 & 0 & {\rho_{118d}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33d} } & {\rho_{109d}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{109d} } & {\rho_{33d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{118d} } & 0 & 0 & {\rho_{11d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{127d} } & 0 & 0 & 0 & 0 & {\rho_{11d} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{136d} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33d} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{145d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11d} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho_{154d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11d} } & 0 & 0 & 0 \\ 0 & 0 & {\rho_{163d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33d} } & 0 & 0 \\ 0 & {\rho_{172d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11d} } & 0 \\ {\rho_{181d} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11d} } \\ \end{array} } \right) $$
$$ \rho_{11d} = \frac{p}{6};\quad \rho_{33d} = \frac{1}{6}\left( {1 - 2p} \right) $$
$$ \rho_{181d} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{3} J_{1} J_{2} J_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} + c_{0} J_{1} + c_{0} J_{2} + c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{172d} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{2} J_{1} J_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + c_{0} J_{1} + c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
$$ \rho_{163d} = \frac{{\left( {1 - 2p} \right)}}{{6\left( {\Delta_{c} t} \right)^{3} J_{1} J_{2} J_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} - \varepsilon_{3} + c_{0} J_{1} + c_{0} J_{2} - c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{154d} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{2} J_{1} J_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{3} + c_{0} J_{1} + c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{145d} = \frac{p}{{\left( {6\Delta_{c} tJ_{1} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + c_{0} J_{1} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right) $$
$$ \rho_{136d} = \frac{1 - 2p}{{6\left( {\Delta_{c} t} \right)^{2} J_{1} J_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{3} + c_{0} J_{1} - c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{127d} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{3} J_{1} J_{2} J_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} + c_{0} J_{1} - c_{0} J_{2} + c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{118d} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{2} J_{1} J_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + c_{0} J_{1} - c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$

\(\rho_{109d} = \frac{{\left( {1 - 2p} \right)}}{{6\left( {\Delta_{c} t} \right)^{3} J_{1} J_{2} J_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} + c_{0} J_{1} - c_{0} J_{2} - c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ_{2} } \right)\sin \left( {\Delta_{c} tJ_{3} } \right)\)

  1. (B2)

    Common environments (ce)

$$ \rho_{11c} = \frac{p}{6};\quad \rho_{33c} = \frac{1}{6}\left( {1 - 2p} \right) $$
$$ \rho_{181c} = \frac{p}{{\left( {18J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} + 3Jc_{0} } \right)t} \right\}\sin \left( {3Jt\Delta_{c} } \right) $$
$$ \rho_{172c} = \frac{p}{{\left( {12J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + 2Jc_{0} } \right)t} \right\}\sin \left( {2Jt\Delta_{c} } \right) $$
$$ \rho_{163c} = \frac{{\left( {1 - 2p} \right)}}{{\left( {6J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} - \varepsilon_{3} + Jc_{0} } \right)t} \right\}\sin \left( {Jt\Delta_{c} } \right) $$
$$ \rho_{154c} = \frac{p}{{\left( {12J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{3} + 2Jc_{0} } \right)t} \right\}\sin \left( {2Jt\Delta_{c} } \right) $$
$$ \rho_{145c} = \frac{p}{{\left( {6J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + Jc_{0} } \right)t} \right\}\sin \left( {Jt\Delta_{c} } \right) $$
$$ \rho_{136c} = \frac{1 - 2p}{6}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{3} } \right)t} \right\} $$
$$ \rho_{127c} = \frac{p}{{\left( {6J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} + Jc_{0} } \right)t} \right\}\sin \left( {Jt\Delta_{c} } \right) $$
$$ \rho_{118c} = \frac{p}{6}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} } \right)t} \right\} $$
$$ \rho_{109c} = \frac{{\left( {1 - 2p} \right)}}{{\left( {6J\Delta_{c} t} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} - Jc_{0} } \right)t} \right\}\sin \left( {Jt\Delta_{c} } \right) $$
  1. (B3)

    Bipartite environments (be)

  1. a—

    The case where \(\left( {Q_{1} } \right)\) and \(\left( {Q_{2} } \right)\) are in common environment.

$$ \rho_{11Bea} = \frac{p}{6};\quad \rho_{33Bea} = \frac{1}{6}\left( {1 - 2p} \right) $$
$$ \rho_{181Bea} = \frac{p}{{12\left( {\Delta_{c} t} \right)^{2} JJ_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} + 2c_{0} J + c_{0} J_{3} } \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{172Bea} = \frac{p}{{\left( {12\Delta_{c} Jt} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + 2c_{0} J} \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right) $$
$$ \rho_{163Ba} = \frac{{\left( {1 - 2p} \right)}}{{12\left( {\Delta_{c} t} \right)^{2} JJ_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} - \varepsilon_{3} + 2c_{0} J - c_{0} J_{3} } \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{154Bea} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{2} JJ_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{3} + c_{0} J + c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{145Bea} = \frac{p}{{\left( {6\Delta_{c} tJ} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ} \right) $$
$$ \rho_{136Bea} = \frac{{\left( {1 - 2p} \right)}}{{6\left( {\Delta_{c} t} \right)^{2} JJ_{3} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{3} + c_{0} J - c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{127Bea} = \frac{p}{{\left( {6\Delta_{c} tJ_{3} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} + c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{3} } \right) $$
$$ \rho_{118Bea} = \frac{p}{6}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} } \right)t} \right\} $$
$$ \rho_{109Bea} = \frac{{\left( {1 - 2p} \right)}}{{\left( {6\Delta_{c} tJ_{3} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} - c_{0} J_{3} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{3} } \right) $$
  1. b—

    The case where \(\left( {Q_{1} } \right)\) and \(\left( {Q_{3} } \right)\) are in common environment.

    $$ \rho_{11Beb} = \frac{p}{6};\quad \rho_{33Beb} = \frac{1}{6}\left( {1 - 2p} \right) $$
    $$ \rho_{181Beb} = \frac{p}{{12\left( {\Delta_{c} t} \right)^{2} JJ_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} + 2c_{0} J + c_{0} J_{2} } \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
    $$ \rho_{172Beb} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{2} JJ_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + c_{0} J + c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
    $$ \rho_{163Beb} = \frac{{\left( {1 - 2p} \right)}}{{\left( {6\Delta_{c} tJ_{2} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} - \varepsilon_{3} + c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{2} } \right) $$
    $$ \rho_{154Beb} = \frac{p}{{\left( {12\Delta_{c} tJ} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{3} + 2c_{0} J} \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right) $$
    $$ \rho_{145Beb} = \frac{p}{{\left( {6\Delta_{c} tJ} \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ} \right) $$
    $$ \rho_{136Beb} = \frac{1 - 2p}{6}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{3} } \right)t} \right\} $$
    $$ \rho_{127Beb} = \frac{p}{{12\left( {\Delta_{c} t} \right)^{2} JJ_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} + 2c_{0} J - c_{0} J_{2} } \right)t} \right\}\sin \left( {2\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
    $$ \rho_{118Beb} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{2} JJ_{2} }}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + c_{0} J - c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ} \right)\sin \left( {\Delta_{c} tJ_{2} } \right) $$
    $$ \rho_{109Beb} = \frac{{\left( {1 - 2p} \right)}}{{\left( {6\Delta_{c} tJ_{2} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} - c_{0} J_{2} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{2} } \right) $$
  1. c—

    The case where \(\left( {Q_{2} } \right)\) and \(\left( {Q_{3} } \right)\) are in common environment.

    $$ \rho_{11Bec} = \frac{p}{6};\quad \rho_{33Bec} = \frac{1}{6}\left( {1 - 2p} \right) $$
    $$ \rho_{181Bec} = \frac{p}{{12\left( {\Delta_{c} t} \right)^{2} JJ_{1} }}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3} + c_{0} J_{1} + 2c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {2\Delta_{c} tJ} \right) $$
    $$ \rho_{172Bec} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{2} J_{1} J}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} + c_{0} J_{1} + c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ} \right) $$
    $$ \rho_{163Bec} = \frac{{\left( {1 - 2p} \right)}}{{\left( {6\Delta_{c} tJ_{1} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{2} - \varepsilon_{3} + c_{0} J_{1} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right) $$
    $$ \rho_{154Bec} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{2} J_{1} J}}\exp \left\{ {2i\left( {\varepsilon_{1} + \varepsilon_{3} + c_{0} J_{1} + c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ} \right) $$
    $$ \rho_{145Bec} = \frac{p}{{\left( {6\Delta_{c} tJ_{1} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} + c_{0} J_{1} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right) $$
    $$ \rho_{136Bec} = \frac{1 - 2p}{{6\left( {\Delta_{c} t} \right)^{2} J_{1} J}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{3} + c_{0} J_{1} - c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ} \right) $$
    $$ \rho_{127Bec} = \frac{p}{{\left( {6\Delta_{c} tJ_{1} } \right)}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} + c_{0} J_{1} } \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right) $$
    $$ \rho_{118Bec} = \frac{p}{{6\left( {\Delta_{c} t} \right)^{2} J_{1} J}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} + c_{0} J_{1} - c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {\Delta_{c} tJ} \right) $$
    $$ \rho_{109Bec} = \frac{{\left( {1 - 2p} \right)}}{{12\left( {\Delta_{c} t} \right)^{2} J_{1} J}}\exp \left\{ {2i\left( {\varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} + c_{0} J_{1} - 2c_{0} J} \right)t} \right\}\sin \left( {\Delta_{c} tJ_{1} } \right)\sin \left( {2\Delta_{c} tJ} \right) $$

Appendix B Explicit derivation of the partial transposed of these matrices

  • First system

The transpose with respect to \(Q_{1}\) is equal to The transpose with respect to \(Q_{3}\).

$$ \rho_{1Q1} = \rho_{1}^{T1} = \left( {\begin{array}{*{20}c} {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{76} } \\ 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{85} } & 0 \\ 0 & 0 & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{94} } & 0 & 0 \\ 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & {\rho_{103} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & {\rho_{112} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & {\rho_{121} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{121}^{ + } } & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{112}^{ + } } & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho_{103}^{ + } } & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 \\ 0 & 0 & {\rho_{94}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & 0 & 0 \\ 0 & {\rho_{85}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 \\ {\rho_{76}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } \\ \end{array} } \right) $$

The transpose with respect to the second Qubit is:

$$ \rho_{1Q2} = \rho_{1}^{T2} = \left( {\begin{array}{*{20}c} {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{94}^{ + } } \\ 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{85}^{ + } } & 0 \\ 0 & 0 & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{76}^{ + } } & 0 & 0 \\ 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & {\rho_{121}^{ + } } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & {\rho_{112}^{ + } } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & {\rho_{103}^{ + } } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{103} } & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{112} } & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho_{121} } & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 \\ 0 & 0 & {\rho_{76} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & 0 & 0 \\ 0 & {\rho_{85} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 \\ {\rho_{94} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } \\ \end{array} } \right) $$

  • Second system

$$ \rho_{2Q1} = \rho_{2}^{T1} = \rho_{2Q3} = \left( {\begin{array}{*{20}c} {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{109} } \\ 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{118} } & 0 \\ 0 & 0 & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{127} } & 0 & 0 \\ 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{136} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{145} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{154} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & {\rho_{163} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & {\rho_{172} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & {\rho_{181} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{181}^{ + } } & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{172}^{ + } } & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{163}^{ + } } & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{154}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{145}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho_{136}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 \\ 0 & 0 & {\rho_{127}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & 0 & 0 \\ 0 & {\rho_{118}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 \\ {\rho_{109}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } \\ \end{array} } \right) $$
$$ \rho_{2Q2} = \rho_{2}^{T2} = \left( {\begin{array}{*{20}c} {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{127}^{ + } } \\ 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{118}^{ + } } & 0 \\ 0 & 0 & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{109}^{ + } } & 0 & 0 \\ 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{154}^{ + } } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{145}^{ + } } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{136}^{ + } } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & {\rho_{181}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & {\rho_{172}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & {\rho_{163}^{ + } } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{163} } & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{172} } & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{181} } & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\rho_{136} } & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\rho_{145} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\rho_{154} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 & 0 & 0 \\ 0 & 0 & {\rho_{109} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{33} } & 0 & 0 \\ 0 & {\rho_{118} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } & 0 \\ {\rho_{127} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\rho_{11} } \\ \end{array} } \right) $$

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Tiokang, O.M., Tchangnwa, F.N., Tchinda, J.D. et al. Comparative study of the dynamics of entanglement and purity in two hybrid quantum systems under the effect of static noise. Eur. Phys. J. Plus 136, 957 (2021). https://doi.org/10.1140/epjp/s13360-021-01950-7

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