Dynamical evolution of entanglement of a three-qubit system driven by a classical environmental colored noise

Abstract

The effects of $1/f^{\alpha } (\alpha =1,2)$ noise stemming from one or a collection of random bistable fluctuators (RBFs), on the evolution of entanglement, of three non-interacting qubits are investigated. Three different initial configurations of the qubits are analyzed in detail: the Greenberger–Horne–Zeilinger (GHZ)-type states, W-type states and mixed states composed of a GHZ state and a W state (GHZ-W). For each initial configuration, the evolution of entanglement is investigated for three different qubit–environment (Q-E) coupling setups, namely independent environments, mixed environments and common environment coupling. With the help of tripartite negativity and suitable entanglement witnesses, we show that the evolution of entanglement is extremely influenced not only by the initial configuration of the qubits, the spectrum of the environment and the Q-E coupling setup considered, but also by the number of RBF modeling the environment. Indeed, we find that the decay of entanglement is accelerated when the number of fluctuators modeling the environment is increased. Furthermore, we find that entanglement can survive indefinitely to the detrimental effects of noise even for increasingly larger numbers of RBFs. On the other hand, we find that the proficiency of the tripartite entanglement witnesses to detect entanglement is weaker than that of the tripartite negativity and that the symmetry of the initial states is broken when the qubits are coupled to the noise in mixed environments. Finally, we find that the 1 / f noise is more harmful to the survival of entanglement than the $1/f^{2}$ noise and that the mixed GHZ-W states followed by the GHZ-type states preserve better entanglement than the W-type ones.

Appendices

Appendix A: Explicit forms of the time-evolved three-qubit density matrix: the case of GHZ-type states

Here, we give the explicit forms of the time-evolved density matrix obtained from Eq. (8) when the three qubits are initially prepared in the GHZ-type states of Eq. (10) and then subjected either to a collection of N RBFs or to a single RBF (N = 1) in common, independent and mixed environments. Throughout this section, we have

$$\begin{aligned} \tau (t)=\dfrac{r\eta _{2}^{2}(t)}{4}=\dfrac{r}{4}\left[ \int \limits _{\,\,\gamma _{\min }}^{\gamma _{\max }}G_{2\lambda }(\gamma ,t){{\mathrm{P}}}(\gamma )\,\gamma \right] ^{2N} \end{aligned}$$

and

$$\begin{aligned} \chi (t)=\dfrac{r}{16}(\eta _{4}(t)+1)=\dfrac{r}{16}\left( \left[ \int \limits _{\,\,\gamma _{\min }}^{\gamma _{\max }}G_{4\lambda }(\gamma ,t){{\mathrm{P}}}(\gamma )\,\mathrm{d}\gamma \right] ^{N}+1\right) . \end{aligned}$$

where the time-dependent function $G_{n\lambda }(\gamma ,t)$ has been already defined in Eq. (21).

1.1 Independent environments (IE)

For this Q-E interaction configuration, we find that the density matrix of the system at a given time t can be written as

$$\begin{aligned} \rho _\mathrm{GHZ}^\mathrm{IE}(t)=\dfrac{1}{8} \left[ \begin{array}{cccccccc} \mathcal {E}(t) &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\mathcal {D}}(t) \\ 0&{} {\mathcal {K}}(t) &{} 0 &{} 0 &{} 0&{} 0 &{} {\mathcal {F}}(t) &{}0 \\ 0 &{} 0 &{} {\mathcal {K}}(t) &{} 0 &{} 0 &{} {\mathcal {F}}(t) &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} {\mathcal {K}}(t) &{} {\mathcal {F}}(t) &{} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} {\mathcal {F}}(t) &{} {\mathcal {K}}(t) &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} {\mathcal {F}}(t) &{} 0 &{} 0 &{} {\mathcal {K}}(t) &{} 0 &{} 0\\ 0&{} {\mathcal {F}}(t) &{} 0 &{} 0 &{} 0 &{} 0 &{} {\mathcal {K}}(t) &{} 0 \\ {\mathcal {D}}(t)&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \mathcal {E}(t) \end{array}\right] , \end{aligned}$$

(A.1)

with $\mathcal {E}(t)=12\tau (t)+1$, ${\mathcal {K}}(t)=1-4\tau (t)$, ${\mathcal {F}}(t)=r-4\tau (t)$, and ${\mathcal {D}}(t)=r+12\tau (t)$.

1.2 Mixed environments (ME)

On the other hand, for the case of ME coupling we find that the dynamics of the system is described by the following density matrix

$$\begin{aligned} \rho _\mathrm{GHZ}^\mathrm{ME}(t)= \left[ \begin{array}{cccccccc} \mathcal {T}(t)+\dfrac{1}{8}&{} -\,\mathcal {G}(t) &{} 0 &{} 0 &{} 0 &{} 0 &{} -\,\mathcal {G}(t) &{} \mathcal {T}(t)+\dfrac{r}{8} \\ -\,\mathcal {G}(t)&{} {\mathcal {Y}}(t)+\dfrac{1}{8} &{} 0 &{} 0 &{} 0 &{} 0 &{} {\mathcal {Y}}(t)+\dfrac{r}{8} &{} -\,\mathcal {G}(t) \\ 0&{} 0 &{} \mathcal {M}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} 0 &{} 0 \\ 0&{} 0 &{} \mathcal {G}(t) &{} \mathcal {M}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} 0 &{} 0\\ 0&{} 0 &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {M}(t) &{} \mathcal {G}(t) &{} 0&{} 0 \\ 0&{} 0 &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {M}(t) &{} 0 &{} 0 \\ -\,\mathcal {G}(t)&{} {\mathcal {Y}}(t)+\dfrac{r}{8} &{} 0 &{} 0 &{} 0 &{} 0 &{} {\mathcal {Y}}(t)+\dfrac{1}{8} &{} -\,\mathcal {G}(t) \\ \mathcal {T}(t)+\dfrac{r}{8} &{} -\,\mathcal {G}(t) &{} 0 &{} 0 &{}0 &{} 0&{} -\,\mathcal {G}(t) &{} \mathcal {T}(t)+\dfrac{1}{8} \end{array}\right] , \end{aligned}$$

(A.2)

where $\mathcal {T}(t)=\chi (t)+\tau (t)$, ${\mathcal {Y}}(t)=\chi (t)-\tau (t)$, $\mathcal {G}(t)=\dfrac{r}{8}-\chi (t)$ and $\mathcal {M}(t)= \dfrac{1}{8}-\chi (t)$

1.3 Common environment (CE)

Finally, for the case of CE coupling, the density matrix describing the evolution of the system results into

$$\begin{aligned} \rho _\mathrm{GHZ}^\mathrm{CE}(t)= \left[ \begin{array}{cccccccc} 3\chi (t)+\dfrac{1}{8}&{} -\,\mathcal {G}(t) &{} -\,\mathcal {G}(t) &{} -\,\mathcal {G}(t) &{} -\,\mathcal {G}(t) &{} -\,\mathcal {G}(t) &{} -\,\mathcal {G}(t) &{} 3\chi (t)+\dfrac{r}{8} \\ -\,\mathcal {G}(t)&{} \mathcal {M}(t) &{}\mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} -\,\mathcal {G}(t) \\ -\,\mathcal {G}(t)&{} \mathcal {G}(t) &{} \mathcal {M}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} -\,\mathcal {G}(t) \\ -\,\mathcal {G}(t)&{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {M}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} -\,\mathcal {G}(t) \\ -\,\mathcal {G}(t)&{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {M}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} -\,\mathcal {G}(t)\\ -\,\mathcal {G}(t)&{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t)&{} \mathcal {M}(t) &{} \mathcal {G}(t) &{} -\,\mathcal {G}(t) \\ -\,\mathcal {G}(t)&{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} \mathcal {M}(t) &{} -\,\mathcal {G}(t)\\ 3\chi (t)+\dfrac{r}{8}&{} -\,\mathcal {G}(t) &{} -\,\mathcal {G}(t) &{} -\,\mathcal {G}(t) &{} -\,\mathcal {G}(t) &{} -\,\mathcal {G}(t) &{} -\,\mathcal {G}(t) &{} 3\chi (t)+\dfrac{1}{8} \end{array}\right] , \end{aligned}$$

(A.3)

where $\mathcal {G}(t)$ and $\mathcal {M}(t)$ have been already defined above.

We observe immediately that the form of the time-evolved density of the system initially prepared in the GHZ-type states depends upon the Q-E coupling configuration considered. This is in good agreement with the results of Ref. [30].

Appendix B: Explicit forms of the time-evolved three-qubit density matrix: the case of W-type states

Here, we give the explicit forms of the time-evolved density matrix obtained from Eq. (8) when the three qubits are initially prepared in the W-type states of Eq. (11) and then subjected either to a collection of N RBFs or to a single RBF (N = 1) in common, independent and mixed environments. Throughout this section, we recall that $\eta _{n}(t)= \left[ \int \limits _{\gamma _{\min }}^{\gamma _{\max }}G_{n\lambda }(\gamma ,t){{\mathrm{P}}}(\gamma )\,\mathrm{d}\gamma \right] ^{2N}$, with $n\in \left\{ 2,4,6 \right\}$.

1.1 Independent environments (IE)

If we set $\mathcal {E}=1-r$, $\mathcal {X}(t)=1+\eta _{2}(t)$ and ${\mathcal {A}}=1-\eta _{2}(t)$, the density matrix for this configuration of the Q-E interaction can be written as

$$\begin{aligned} \rho _{W}^\mathrm{IE}(t)=\dfrac{1}{8} \left[ \begin{array}{cccccccc} \mathcal {L}(t)&{} 0 &{} 0 &{} \dfrac{4\mathcal {L}(t)}{3} &{} 0 &{} \dfrac{4\mathcal {L}(t)}{3} &{} \dfrac{4\mathcal {L}(t)}{3} &{} 0 \\ 0&{} {\mathcal {Z}}(t) &{} {\mathcal {K}}(t) &{} 0 &{} {\mathcal {K}}(t) &{} 0 &{} 0 &{} \dfrac{4{\mathcal {B}}(t)}{3} \\ 0&{} {\mathcal {K}}(t) &{} {\mathcal {Z}}(t) &{} 0 &{} {\mathcal {K}}(t) &{} 0 &{} 0 &{} \dfrac{4{\mathcal {B}}(t)}{3} \\ \dfrac{4\mathcal {L}(t)}{3}&{} 0 &{} 0 &{} {\mathcal {I}}(t) &{} 0 &{} \mathcal {O}(t) &{} \mathcal {O}(t) &{} 0 \\ 0&{} {\mathcal {K}}(t) &{} {\mathcal {K}}(t) &{} 0 &{} {\mathcal {Z}}(t) &{} 0 &{} 0 &{} \dfrac{4{\mathcal {B}}(t)}{3}\\ \dfrac{4\mathcal {L}(t)}{3}&{} 0 &{} 0 &{} \mathcal {O}(t) &{} 0 &{} {\mathcal {I}}(t) &{} \mathcal {O}(t) &{} 0 \\ \dfrac{4\mathcal {L}(t)}{3}&{} 0 &{} 0 &{} \mathcal {O}(t) &{} 0 &{} \mathcal {O}(t) &{} {\mathcal {I}}(t) &{} 0 \\ 0&{} \dfrac{4{\mathcal {B}}(t)}{3} &{} \dfrac{4{\mathcal {B}}(t)}{3} &{} 0 &{} \dfrac{4{\mathcal {B}}(t)}{3} &{} 0 &{} 0 &{} {\mathcal {B}}(t) \end{array}\right] , \end{aligned}$$

(B.1)

where

$$\begin{aligned} \mathcal {L}(t)= & {} r\mathcal {X}^{2}(t){\mathcal {A}}(t)+\mathcal {E},\\ {\mathcal {B}}(t)= & {} r\mathcal {X}(t){\mathcal {A}}^{2}(t)+\mathcal {E}, \\ {\mathcal {Z}}(t)= & {} \dfrac{r\mathcal {X}(t)}{3}\left[ \mathcal {X}^{2}(t)+2{\mathcal {A}}^{2}(t)\right] +\mathcal {E},\\ {\mathcal {I}}(t)= & {} \dfrac{r{\mathcal {A}}(t)}{3}\left[ {\mathcal {A}}^{2}(t)+2\mathcal {X}^{2}(t)\right] +\mathcal {E},\\ {\mathcal {K}}(t)= & {} \dfrac{2r\mathcal {X}(t)}{3}\left[ \mathcal {X}^{2}(t)-\mathcal {X}(t)+{\mathcal {A}}(t)\right] \\&\hbox { and } \mathcal {O}(t)=\dfrac{2r{\mathcal {A}}(t)}{3}\left[ \mathcal {X}^{2}(t)-\mathcal {X}(t)+{\mathcal {A}}(t)\right] . \end{aligned}$$

1.2 Mixed environments (ME)

On the other hand, when the subsystems are coupled in mixed environments, the time-evolved density matrix of the system takes the form

$$\begin{aligned} \rho _{W}^\mathrm{ME}(t)= \left[ \begin{array}{cccccccc} {\mathcal {Z}}(t)&{} 0 &{} 0 &{} {\mathcal {Y}}(t) &{} 0&{} {\mathcal {Y}}(t) &{} {\mathcal {Y}}(t) &{} 0 \\ 0 &{}\mathcal {X}(t) &{} {\mathcal {J}}(t) &{} 0 &{} \mathcal {Q}(t) &{} 0&{} 0 &{} {\mathcal {V}}(t) \\ 0&{} {\mathcal {J}}(t) &{}\mathcal {X}(t) &{} 0 &{} \mathcal {Q}(t) &{} 0 &{} 0 &{} {\mathcal {V}}(t) \\ {\mathcal {Y}}(t) &{} 0 &{} 0 &{} \mathcal {O}(t) &{} 0 &{} \mathcal {T}(t) &{} \mathcal {T}(t) &{} 0 \\ 0 &{} \mathcal {Q}(t) &{} \mathcal {Q}(t) &{} 0&{} {\mathcal {K}}(t) &{} 0 &{} 0 &{} \mathcal {L}(t) \\ {\mathcal {Y}}(t)&{} 0 &{} 0 &{} \mathcal {T}(t) &{} 0 &{} \mathcal {O}(t) &{} \mathcal {G}(t) &{} 0 \\ {\mathcal {Y}}(t)&{} 0 &{} 0 &{} \mathcal {T}(t) &{} 0 &{} \mathcal {G}(t) &{} \mathcal {O}(t) &{} 0 \\ 0 &{} {\mathcal {V}}(t) &{} {\mathcal {V}}(t) &{} 0&{} \mathcal {L}(t) &{} 0 &{} 0 &{} \mathcal {P}(t) \end{array}\right] , \end{aligned}$$

(B.2)

where

$$\begin{aligned} \mathcal {X}(t)= & {} \dfrac{r}{48}+\dfrac{1}{8}+r\left\{ \dfrac{1}{12}\eta _{2}^{2}(t)+\dfrac{3}{48}\eta _{2}(t)-\dfrac{1}{16}\eta _{4}(t)+\dfrac{10}{96}\eta _{2}(t)\eta _{4}(t)\right\} ,\\ \mathcal {O}(t)= & {} -\dfrac{r}{48}+\dfrac{1}{8}+r\left\{ -\dfrac{1}{16}\eta _{2}(t)+\dfrac{1}{16}\eta _{4}(t)-\dfrac{10}{96}\eta _{2}(t)\eta _{4}(t)\right\} , \\ {\mathcal {J}}(t)= & {} \dfrac{r}{12}\left\{ \eta _{2}^{2}(t)+\eta _{2}(t)+\eta _{2}(t)\eta _{4}(t)+1 \right\} , \\ {\mathcal {Z}}(t)= & {} \dfrac{r}{48}+\dfrac{1}{8}+r\left\{ -\dfrac{1}{12}\eta _{2}^{2}(t)+\dfrac{5}{48}\eta _{2}(t)-\dfrac{1}{16}\eta _{4}(t)-\dfrac{10}{96}\eta _{2}(t)\eta _{4}(t)\right\} ,\\ \mathcal {P}(t)= & {} \dfrac{r}{48}+\dfrac{1}{8}+r\left\{ -\dfrac{1}{12}\eta _{2}^{2}(t)-\dfrac{5}{48}\eta _{2}(t)-\dfrac{1}{16}\eta _{4}(t)+\dfrac{10}{96}\eta _{2}(t)\eta _{4}(t)\right\} , \\ {\mathcal {K}}(t)= & {} -\,\dfrac{r}{48}+\dfrac{1}{8}+r\left\{ \dfrac{1}{16}\eta _{2}(t)+\dfrac{1}{16}\eta _{4}(t)+\dfrac{10}{96}\eta _{2}(t)\eta _{4}(t)\right\} , \\ {\mathcal {V}}(t)= & {} -\dfrac{r}{96}\left\{ 10\eta _{2}(t)+6\eta _{4}(t)-10\eta _{2}(t)\eta _{4}(t)-6\right\} , \\ \mathcal {L}(t)= & {} \dfrac{r}{12}\left\{ -\eta _{2}^{2}(t)-\eta _{2}(t)+\eta _{2}(t)\eta _{4}(t)+1 \right\} , \\ \mathcal {T}(t)= & {} \dfrac{r}{96}\left\{ 6\eta _{4}(t)-6\eta _{2}(t)-10\eta _{2}(t)\eta _{4}(t)+10 \right\} , \\ {\mathcal {Y}}(t)= & {} -\,\dfrac{r}{12}\left\{ \eta _{2}^{2}(t)-\eta _{2}(t)+\eta _{2}(t)\eta _{4}(t)-1 \right\} , \\ \mathcal {G}(t)= & {} \dfrac{r}{12}\left\{ -\eta _{2}^{2}(t)+\eta _{2}(t)+\eta _{2}(t)\eta _{4}(t)-1 \right\} \hbox {and}\\ \mathcal {Q}(t)= & {} \dfrac{r}{12}\left\{ \eta _{2}^{2}(t)+\eta _{2}(t)+\eta _{2}(t)\eta _{4}(t)+1 \right\} . \end{aligned}$$

1.3 Common environment (CE)

Finally, for GHZ input configuration, the density matrix of the system at time t takes the following form when the qubits are coupled in a common environment

$$\begin{aligned} \rho _{W}^{CE}(t)= \left[ \begin{array}{cccccccc} {\mathcal {Z}}(t)&{} 0 &{} 0 &{} \mathcal {T}(t) &{} 0 &{} \mathcal {T}(t) &{} \mathcal {T}(t) &{} 0 \\ 0 &{} \mathcal {X}(t) &{} {\mathcal {D}}(t) &{} 0 &{} {\mathcal {D}}(t) &{} 0 &{} 0 &{} \mathcal {G}(t) \\ 0 &{} {\mathcal {D}}(t) &{} \mathcal {X}(t) &{} 0 &{} {\mathcal {D}}(t) &{} 0 &{} 0 &{} \mathcal {G}(t)\\ \mathcal {T}(t) &{} 0 &{} 0 &{} \mathcal {L}(t) &{} 0 &{} \mathcal {P}(t) &{} \mathcal {P}(t) &{} 0 \\ 0&{} {\mathcal {D}}(t) &{} {\mathcal {D}}(t) &{} 0 &{} \mathcal {X}(t) &{} 0 &{} 0 &{} \mathcal {G}(t) \\ \mathcal {T}(t) &{} 0 &{} 0&{} \mathcal {P}(t) &{} 0 &{} \mathcal {L}(t) &{} \mathcal {P}(t) &{} 0 \\ \mathcal {T}(t)&{} 0 &{} 0 &{} \mathcal {P}(t) &{} 0 &{} \mathcal {P}(t) &{} \mathcal {L}(t) &{} 0 \\ 0&{} \mathcal {G}(t) &{} \mathcal {G}(t) &{} 0 &{} \mathcal {G}(t) &{} 0 &{} 0 &{} \mathcal {O}(t) \end{array}\right] , \end{aligned}$$

(B.3)

where

$$\begin{aligned} {\mathcal {Z}}(t)= & {} \dfrac{1}{8}+r\left\{ \dfrac{1}{16}+\dfrac{3}{32}\eta _{2}(t)-\dfrac{3}{16}\eta _{4}(t)-\dfrac{3}{32}\eta _{6}(t)\right\} ,\\ \mathcal {T}(t)= & {} \dfrac{r}{96}+r\left\{ 9\eta _{2}(t)-6\eta _{4}(t)-9\eta _{6}(t)+6\right\} ,\\ \mathcal {L}(t)= & {} \dfrac{1}{8}-\dfrac{r}{48}\left\{ -\dfrac{7}{96}\eta _{2}(t)+\dfrac{1}{16}\eta _{4}(t)-\dfrac{3}{32}\eta _{6}(t)\right\} ,\\ {\mathcal {D}}(t)= & {} \dfrac{r}{96}\left\{ 7\eta _{2}(t)+6\eta _{4}(t)+9\eta _{6}(t)+10\right\} , \\ \mathcal {P}(t)= & {} \dfrac{r}{96}\left\{ 10-7\eta _{2}(t)+6\eta _{4}(t)-9\eta _{6}(t)\right\} ,\\ \mathcal {O}(t)= & {} \dfrac{1}{8}+\dfrac{r}{16}+r\left\{ -\dfrac{3}{32}\eta _{2}(t)-\dfrac{3}{16}\eta _{4}(t)+\dfrac{3}{32}\eta _{6}(t)\right\} , \\ \mathcal {X}(t)= & {} \dfrac{1}{8}-\dfrac{r}{48}\left\{ \dfrac{7}{96}\eta _{2}(t)+\dfrac{1}{16}\eta _{4}(t)+\dfrac{3}{32} \eta _{6}(t)\right\} \\&\hbox {and } \mathcal {G}(t)=\dfrac{r}{96}\left\{ 6- 9\eta _{2}(t)-6\eta _{4}(t)+9\eta _{6}(t)\right\} . \end{aligned}$$

Appendix C: Explicit forms of the time-evolved three-qubit density matrix: the case of mixed states of Eq. (12)

Here, we give the explicit forms of the time-evolved density matrix obtained from Eq. (8) when the three qubits are initially prepared in the mixed states composed of a W state and a GHZ state of Eq. (12) and then subjected either to a collection of N RBFs or to a single RBF (N = 1) in common, independent and mixed environments. Once more, we recall that throughout this section, the function $\eta _{n}(t)$ with $n\in \left\{ 2,4,6\right\}$ is defined in Eq. (23).

1.1 Independent environments (IE)

In this Q-E configuration, the dynamics of the system is governed by the following density matrix

$$\begin{aligned} \rho _{W-\mathrm{GHZ}}^\mathrm{IE}(t)= \left[ \begin{array}{cccccccc} {\mathcal {I}}_{1}(t)&{} 0 &{} 0 &{} {\mathcal {Y}}(t) &{} 0 &{} {\mathcal {Y}}(t) &{} {\mathcal {Y}} (t)&{} -\,3\mathcal {G}(t)+4r \\ 0&{} {\mathcal {I}}_{2}(t) &{} \mathcal {R}(t) &{} 0 &{} \mathcal {R}(t) &{} 0 &{} \mathcal {G}(t) &{} {\mathcal {F}}(t) \\ 0&{} \mathcal {R}(t) &{} {\mathcal {I}}_{2}(t) &{} 0 &{} \mathcal {R}(t) &{} \mathcal {G}(t) &{} 0 &{} {\mathcal {F}}(t) \\ {\mathcal {Y}}(t)&{} 0 &{} 0 &{} {\mathcal {I}}_{3}(t)&{} \mathcal {G}(t) &{} \mathcal {C}(t) &{}\mathcal {C}(t) &{} 0\\ 0&{} \mathcal {R}(t) &{} \mathcal {R}(t)&{} \mathcal {G}(t) &{} {\mathcal {I}}_{2}(t) &{} 0 &{} 0 &{} {\mathcal {F}}(t)\\ {\mathcal {Y}}(t)&{} 0 &{} \mathcal {G}(t) &{}\mathcal {C}(t)&{} 0 &{} {\mathcal {I}}_{3}(t) &{} \mathcal {C}(t) &{} 0\\ {\mathcal {Y}}(t) &{} \mathcal {G}(t) &{} 0 &{} \mathcal {C}(t) &{} 0 &{} \mathcal {C}(t) &{} {\mathcal {I}}_{3}(t)&{} 0\\ -\,3\mathcal {G}(t)+4r&{} {\mathcal {F}}(t) &{} {\mathcal {F}}(t) &{} 0 &{} {\mathcal {F}}(t) &{} 0 &{} 0 &{} {\mathcal {I}}_{4}(t) \end{array}\right] , \end{aligned}$$

(C.1)

where

$$\begin{aligned} \mathcal {X}(t)= & {} 1+\eta _{2}(t),{\mathcal {A}}(t)=1-\eta _{2}(t), \mathcal {O}(t)=(p-1)\dfrac{\left[ \eta _{2}^{3}(t)-\eta _{2}(t)\right] }{8}, \\ \mathcal {X}_{1}(t)= & {} \dfrac{(4p-1)\eta _{2}^{2}(t)+1}{8}, \mathcal {X}_{2}(t)=\dfrac{(1-4p)}{24}\eta _{2}^{2}(t)+\dfrac{1}{8}, \\ {\mathcal {D}}_{1}(t)= & {} \dfrac{(1-p){\mathcal {A}}(t)}{12}, {\mathcal {D}}_{2}(t)=\dfrac{(1-p)\mathcal {X}(t)}{12}, \\ \mathcal {G}(t)= & {} \dfrac{p(1-\eta _{2}^{2}(t))}{8}, \mathcal {L}(t)=\dfrac{(1-p)}{8}\left[ \eta _{2}^{3}(t)+\dfrac{1}{3}\eta _{2}(t)\right] , \\ {\mathcal {I}}_{1}(t)= & {} \mathcal {O}(t)+\mathcal {X}_{1}(t), {\mathcal {I}}_{2}(t)=\mathcal {L}(t)+\mathcal {X}_{2}(t), \\ {\mathcal {I}}_{3}(t)= & {} \mathcal {X}_{2}(t)-\mathcal {L}(t), {\mathcal {I}}_{4}(t)=\mathcal {X}_{1}(t)-\mathcal {O}(t),\\ \mathcal {P}(t)= & {} \mathcal {G}(t)-1, {\mathcal {Y}}(t)={\mathcal {D}}_{1}(t)\mathcal {X}^{2}(t),\\ \mathcal {R}(t)= & {} {\mathcal {D}}_{2}(t)\mathcal {P}(t), {\mathcal {F}}(t)={\mathcal {D}}_{2}(t){\mathcal {A}}^{2}(t)\;\hbox {and}\; \mathcal {C}(t)={\mathcal {D}}_{1}(t)\mathcal {P}(t). \end{aligned}$$

1.2 Mixed environments (ME)

Here, the time evolution of the system is governed by the following density matrix

$$\begin{aligned} \rho _{\mathrm{GHZ}-W}^\mathrm{ME}(t)= \left[ \begin{array}{cccccccc} \mathcal {R}(t) &{} \mathcal {E}(t) &{} 0 &{}\mathcal {X}(t) &{} 0 &{} \mathcal {X}(t) &{} \mathcal {M}(t) &{} {\mathcal {A}}(t) \\ \mathcal {E}(t)&{} \mathcal {T}(t) &{} {\mathcal {Z}}(t) &{} 0 &{} {\mathcal {Z}}(t) &{} 0&{} {\mathcal {I}}(t) &{} {\mathcal {Y}}(t)\\ 0&{} {\mathcal {Z}}(t) &{} {\mathcal {K}}(t) &{} \mathcal {Q}(t) &{} {\mathcal {K}}(t) &{} \mathcal {Q}(t) &{} 0 &{} {\mathcal {V}}(t) \\ \mathcal {X}(t)&{} 0 &{} \mathcal {Q}(t) &{} \mathcal {L}(t) &{} \mathcal {Q}(t) &{} \mathcal {L}(t) &{} \mathcal {C}(t)&{} 0 \\ 0&{} {\mathcal {Z}}(t) &{} {\mathcal {K}}(t) &{} \mathcal {Q}(t) &{} {\mathcal {K}}(t)&{} \mathcal {Q}(t) &{} 0 &{} {\mathcal {V}}(t) \\ \mathcal {X}(t)&{} 0 &{} \mathcal {Q}(t) &{} \mathcal {L}(t) &{} \mathcal {Q}(t) &{} \mathcal {L}(t) &{} \mathcal {C}(t) &{} 0 \\ \mathcal {M}(t)&{} {\mathcal {I}}(t) &{} 0 &{} \mathcal {C}(t) &{} 0 &{} \mathcal {C}(t) &{} {\mathcal {J}}(t) &{} \mathcal {E}(t) \\ {\mathcal {A}}(t)&{} {\mathcal {Y}}(t) &{} {\mathcal {V}}(t) &{} 0 &{} {\mathcal {V}}(t) &{} 0 &{} \mathcal {E}(t) &{} {\mathcal {B}}(t) \end{array}\right] , \end{aligned}$$

(C.2)

where

$$\begin{aligned} \mathcal {R}(t)= & {} \dfrac{1}{96}\left[ 8(4p-1)\eta _{2}^{2}(t)+10(p-1)\eta _{2}(t)(\eta _{4}(t)-1)+6(2p-1)\eta _{4}(t)\right] +\dfrac{p}{24}+\dfrac{7}{48},\\ \mathcal {T}(t)= & {} \dfrac{1}{96}\left[ 8(1-4p)\eta _{2}^{2}(t)+10(1-p)\eta _{2}(t)\eta _{4}(t)+6(1-1p)\eta _{2}(t)+6(2p-1)\eta _{4}(t)\right] +\dfrac{p}{24}+\dfrac{7}{48},\\ {\mathcal {J}}(t)= & {} \dfrac{1}{96}\left[ 8(1-4p)\eta _{2}^{2}(t)+10(1p-1)\eta _{2}(t)\eta _{4}(t)+6(p-1)\eta _{2}(t)+6(2p-1)\eta _{4}(t)\right] +\dfrac{p}{24}+\dfrac{7}{48},\\ {\mathcal {Y}}(t)= & {} \dfrac{1}{96}\left[ 10(1-p)\eta _{4}(t)\eta _{2}(t)+6(2p-1)\eta _{4}(t)+10(p-1)\eta _{2}(t)\right] +\dfrac{1}{16}(1-2p),\\ {\mathcal {I}}(t)= & {} -\dfrac{p}{4}\eta _{2}^{2}(t)+\dfrac{p}{16}(\eta _{4}(t)+3),\,\,\mathcal {E}(t)=\dfrac{p}{16}(\eta _{4}(t)-1),\,\,\mathcal {Q}(t)=\dfrac{p}{16}(1-\eta _{4}(t)),\\ {\mathcal {K}}(t)= & {} \dfrac{1}{96}\left[ 10(1-p)\eta _{4}(t)\eta _{2}(t)+6(1-p)\eta _{2}(t)+6(1-2p)\eta _{4}(t)\right] +\dfrac{1}{48}(5-2p),\\ \mathcal {L}(t)= & {} \dfrac{1}{96}\left[ 10(p-1)\eta _{4}(t)\eta _{2}(t)+6(p-1)\eta _{2}(t)+6(1-2p)\eta _{4}(t)\right] +\dfrac{1}{48}(5-2p),\\ {\mathcal {B}}(t)= & {} \dfrac{1}{96}\left[ 8(4p-1)\eta _{2}^{2}(t)+10(p-1)\eta _{2}(t)(1-\eta _{4}(t))+6(2p-1)\eta _{4}(t)\right] +\dfrac{p}{24}+\dfrac{7}{48},\\ {\mathcal {Z}}(t)= & {} \dfrac{(1-p)}{12}\left[ \eta _{2}^{2}(t)+\eta _{4}(t)\eta _{2}(t)+\eta _{2}(t)+1\right] ,\,\,{\mathcal {A}}(t)=\dfrac{p}{4}\eta _{2}^{2}(t)+\dfrac{p}{16}(\eta _{4}(t)+3),\\ \mathcal {M}(t)= & {} \dfrac{1}{96}\left[ 10(p-1)\eta _{4}(t)\eta _{2}(t)+6(2p-1)\eta _{4}(t)+10(1-p)\eta _{2}(t)\right] +\dfrac{1}{16}(1-2p),\\ {\mathcal {V}}(t)= & {} \dfrac{1}{12}\left[ (p-1)\eta _{2}^{2}(t)+(1-p)\eta _{2}(t)\eta _{4}(t)+(p-1)(\eta _{2}(t)-1)\right] ,\\ \mathcal {C}(t)= & {} \dfrac{1}{12}\left[ (1-p)\eta _{2}^{2}(t)+(p-1)\eta _{2}(t)\eta _{4}(t)+(p-1)(\eta _{2}(t)-1)\right] ,\\ \mathcal {X}(t)= & {} \dfrac{1}{12}\left[ (p-1)\eta _{2}^{2}(t)+(p-1)\eta _{2}(t)\eta _{4}(t)+(1-p)(\eta _{2}(t)+1)\right] . \end{aligned}$$

1.3 Common environment (CE)

Finally, when the qubits are embedded in a common environment, the dynamics is governed by the following density matrix

$$\begin{aligned} \rho _{GHZ-W}^{CE}(t)= \left[ \begin{array}{cccccccc} {\mathcal {J}}(t)&{} {\mathcal {V}}(t) &{} {\mathcal {V}}(t) &{} {\mathcal {K}}(t) &{} {\mathcal {V}}(t) &{} {\mathcal {K}}(t) &{} {\mathcal {K}}(t) &{} {\mathcal {Z}}(t) \\ {\mathcal {V}}(t)&{} {\mathcal {Y}}(t) &{} {\mathcal {Y}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {Y}}(t) &{} -\,{\mathcal {V}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {I}}(t) \\ {\mathcal {V}}(t)&{} {\mathcal {Y}}(t) &{} {\mathcal {Y}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {Y}}(t) &{} -\,{\mathcal {V}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {I}}(t) \\ {\mathcal {K}}(t)&{} -\,{\mathcal {V}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {F}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {F}}(t) &{} {\mathcal {F}}(t) &{} {\mathcal {V}}(t) \\ {\mathcal {V}}(t)&{} {\mathcal {Y}}(t) &{} {\mathcal {Y}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {Y}}(t) &{} -\,{\mathcal {V}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {I}}(t) \\ {\mathcal {K}}(t) &{} -\,{\mathcal {V}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {F}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {F}}(t) &{} {\mathcal {F}}(t) &{} {\mathcal {V}}(t) \\ {\mathcal {K}}(t)&{} -\,{\mathcal {V}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {F}}(t) &{} -\,{\mathcal {V}}(t) &{} {\mathcal {F}}(t) &{} {\mathcal {F}}(t) &{} {\mathcal {V}}(t) \\ {\mathcal {Z}}(t)&{} {\mathcal {I}}(t) &{} {\mathcal {I}}(t) &{} {\mathcal {V}}(t) &{} {\mathcal {I}}(t) &{} {\mathcal {V}}(t) &{} {\mathcal {V}}(t) &{} {\mathcal {B}}(t) \end{array}\right] , \end{aligned}$$

(C.3)

with

$$\begin{aligned} {\mathcal {J}}(t)= & {} \dfrac{1}{32}\left[ 3(1-p)\eta _{2}(t)+6(2p-1)\eta _{4}(t)+3(p-1)\eta _{6}(t)\right] +\dfrac{1}{16}(1+2p),\\ {\mathcal {B}}(t)= & {} \dfrac{1}{32}\left[ 3(p-1)\eta _{2}(t)+6(2p-1)\eta _{4}(t)+3(1-p)\eta _{6}(t)\right] +\dfrac{1}{16}(1+2p),\\ {\mathcal {Y}}(t)= & {} \dfrac{1}{96}\left[ 7(1-p)\eta _{2}(t)+6(1-2p)\eta _{4}(t)+9(1-p)\eta _{6}(t)\right] +\dfrac{1}{48}(5-2p),\\ {\mathcal {F}}(t)= & {} \dfrac{1}{96}\left[ 7(p-1)\eta _{2}(t)+6(1-2p)\eta _{4}(t)+9(p-1)\eta _{6}(t)\right] +\dfrac{1}{48}(5-2p),\\ {\mathcal {K}}(t)= & {} \dfrac{1}{32}\left[ 3(1-p)\eta _{2}(t)+2(2p-1)\eta _{4}(t)+3(p-1)\eta _{6}(t)\right] +\dfrac{1}{16}(1-2p),\\ {\mathcal {I}}(t)= & {} \dfrac{1}{32}\left[ 3(p-1)\eta _{2}(t)+2(2p-1)\eta _{4}(t)+3(1-p)\eta _{6}(t)\right] +\dfrac{1}{16}(1-2p),\\ {\mathcal {V}}(t)= & {} \dfrac{p}{16}(\eta _{4}(t)-1)\,\, \text {and}\,\,{\mathcal {Z}}(t)=\dfrac{p}{16}(3\eta _{4}(t)+5). \end{aligned}$$