relation of Eq. (\[antic\]) to exchange the order between $(-iY_1)$ and the exponentials in the above equation, leading to $$\begin{aligned}
U(T) & = & (-iY_1)^{N} e^{-i[H_{0}+(-1)^{N} H'](T-T_N)}  \nonumber \\
& &\ \times\ e^{-i[H_{0}+(-1)^{N-1}H'](T_N-T_{N-1})}  \nonumber \\
& &\ \cdots \nonumber \\
 & &\ \times\  e^{-i[H_{0}+H'](T_3-T_2)}  \nonumber \\
& &\ \times\ e^{-i[H_{0}-H'](T_2-T_1)} \nonumber \\
& &\ \times\ e^{-i[H_{0}+H']T_1} \nonumber \\
&=& (-iY_1)^{N}e^{-iH_0 T} \mathfrak{J}\left[ e^{-i\int_{0}^{T}
F_{N}(t)H_{I}'(t)\mathrm{d}t}\right] \nonumber
 \\
 &\equiv & (-iY_1)^{N} {\cal U}^{(N)}_{+}(T). \label{un}
  \end{aligned}$$ Here $F_{N}(t)$ is already defined in Eq. (\[fn\]), the second equality is obtained by using the interaction representation, with $H_{I}'(t)\equiv e^{iH_0 t}H_I e^{-iH_0 t}$, and the last line defines the operator ${\cal U}^{(N)}_{+}(T)$. Clearly, ${\cal U}^{(N)}_{+}$ is exactly in the form of ${U}^{(N)}_{+}$ defined in Eqs. (\[oldun\]) and (\[upm\]), with $H_0$ replacing $C$ and $H'$ replacing $Z$. This observation motivates us to define $$\begin{aligned}
{\cal U}^{(N)}_{-}(T) \equiv  e^{-iH_0 T} \mathfrak{J}\left[ e^{-i\int_{0}^{T}-
       F_{N}(t)H_{I}'(t)\mathrm{d}t}\right],\end{aligned}$$ which is completely in parallel with ${U}^{(N)}_{-}$ defined in Eq. (\[upm\]). As such, Eq. (\[Tn1\]) directly leads to $$\begin{aligned}
\left({\cal U}^{(N)}_{-}\right)^{\dagger} {\cal U}^{(N)}_{+} = 1+O(T^{N+1}).
\label{ourTn1}\end{aligned}$$ With Eq. (\[ourTn1\]) obtained we can now evaluate the coherence measure. In particular, for an arbitrary initial state given by the density operator $\rho_{i}$, the expectation value of ${\cal
