Critical assessment of two-qubit post-Markovian master equations

A post-Markovian master equation has been recently proposed as a tool to describe the evolution of a system coupled to a memory-keeping environment [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(R) (2005)]. For a single qubit affected by appropriately chosen environmental conditions, the corresponding dynamics is always legitimate and physical. Here we extend such a situation to the case of two qubits, only one of which experiences the environmental effects. We show how, despite the innocence of such an extension, the introduction of the second qubit should be done cum grano salis to avoid consequences such as the breaking of the positivity of the associated dynamical map. This hints at the necessity of using care when adopting phenomenologically derived models for evolutions occurring outside the Markovian framework.

Figure 1

(a) Trace distance $\mathcal{T}$ (dashed line) and lowest eigenvalue ${\varepsilon }_l$ (solid line) of the mapped matrix $\mathit{\rho }\left(t\right)$ solution of Eq. (4) against the rescaled interaction time $\tilde{t}=Jt$ for the initial state $\mathit{\rho }\left(0\right)={|0\rangle }_S\otimes \left(\right|0\rangle +{|1\rangle )}_A/\sqrt{2}$ and $J=2$. (b) Lowest eigenvalue ${\varepsilon }_l$ of the matrix $\mathit{\rho }\left(t\right)$ as a function of rescaled time $\tau =\gamma t$ and the ratio $\mathcal{R}=J/\gamma$ for the same initial state of panel (a). (c) Plot of the lowest eigenvalue ${\varepsilon }_{cl}$ of the Choi matrix associated with the dynamics of $\mathit{\rho }\left(t\right)$. (d) Lowest eigenvalue ${\varepsilon }_l$ of the matrix $\mathit{\rho }\left(t\right)$ solving Eq. (7). In all the panels we have taken $\overline{n}=1$ with $\chi =\gamma$.