A measure of non-Markovianity for unital quantum dynamical maps

Abstract

One of the most important topics in the study of the dynamics of open quantum systems is the information exchange between system and environment. Based on the features of back-flow information from an environment to a system, an approach is provided to detect non-Markovianity for unital dynamical maps. The method takes advantage of non-contraction property of the von Neumann entropy under completely positive and trace-preserving unital maps. Accordingly, for the dynamics of a single qubit as an open quantum system, the sign of the time derivative of the density matrix eigenvalues of the system determines the non-Markovianity of unital quantum dynamical maps. The main characteristics of the measure are to make the corresponding calculations and optimization procedure simpler.

Appendix

The pure dephasing interaction between a two-level system and surrounding bosonic environment is given by

$$\begin{aligned} H=\frac{\omega _{0}}{2}\sigma _{z}+\sum _{k}\omega _{k}b_{k}^{\dag }b_{k} +\sum _{k}\sigma _{k}\left( g_{k}b_{k}^{\dag }+g_{k}^{*}b_{k}\right) , \end{aligned}$$

(36)

where $\sigma _{z}$ is the usual Pauli matrix in the z-direction, $\omega _{0}$ is the two-level system frequency, the $b_{k}$($b_{k}^{\dag }$) are the annihilation (creation) operators which satisfy the commutation relation $\left[ b_{k},b^{\dag }_{\acute{k}}\right] =\delta _{k,\acute{k}}$, $g_{k}$ is a constant which can control the strength of the coupling between the system and the environment. In this model, the off-diagonal elements of the density matrix of the two-level system decay during the quantum process, while the diagonal elements are constant in time because there is no transition between energy levels which is due to this fact that $[H,\sigma _{z}]=0$ holds. In interaction picture, the Hamiltonian is obtained as

$$\begin{aligned} H_{I}(t)=\sum _{K}\sigma _{z}\left( g_{k}a_{k}^{\dag }\hbox {e}^{i\omega _{k}t} +g_{k}^{*}a_{k}\hbox {e}^{-\omega _{k}t}\right) . \end{aligned}$$

(37)

When the system interacts with a large environment, one can work in the continuum limit and have a replacement $\sum _{k} \vert g_{k} \vert ^{2}\longrightarrow \int d\omega J(\omega ) \delta (\omega _{k}-\omega )$, where $J(\omega )$ is the spectral density of the environment [34, 35]. Using the second-order time-convolutionless master equation [1, 2] at zero temperature, one can find the master equation appeared in Eq. (24) . This model can be described in the Kraus representation form as

$$\begin{aligned} \rho ^{\mathcal {S}}(t)=\sum _{i=1}^{2} D_{i}(t)\rho ^{\mathcal {S}}(0)D_{i}^{\dag }(t), \end{aligned}$$

(38)

where the Kraus operators $D_{i}(t)$ are given by

$$\begin{aligned} D_{1}(t)=\sqrt{\frac{1+\hbox {e}^{-\varGamma (t)}}{2}} \, I, \quad D_{2}(t)=\sqrt{\frac{1-\hbox {e}^{-\varGamma (t)}}{2}}\, \sigma _{z}. \end{aligned}$$

(39)