Quantum discord and non-Markovianity of quantum dynamics

The problem of recognizing (non-)Markovianity of a quantum dynamics is revisited through analyzing quantum correlations. We argue that instantaneouslyvanishing quantum discord provides a necessary and sufficient condition for Markovianity of a quantum map. This is used to introduce a measure of non-Markovianity. This measure, however, requires demanding knowledge about the system and the environment. By using a quantum correlation monogamy property and an ancillary system, we propose a simplified measure with fewer requirements. Non-Markovianity is thereby decided by quantum state tomography of the system and the ancilla.

Figure 1

The initial state of the system ($S$) and environment ($E$) is a generally mixed state with vanishing discord. The state of the ancilla ($A$) is chosen such that it purifies ${\varrho }_{SE}\left(0\right)$; i.e., ${\varrho }_{SEA}\left(0\right)={|\psi \left(0\right)\rangle }_{SEA}\langle \psi \left(0\right)|$. The state of the combined system ($SEA$) hence will remain pure in time. After state tomography to determine ${\varrho }_{\text{SA}}\left(t\right)$, we are able to conclude whether the map $\widehat{\mathcal{E}}$ (acting only on the system) is Markovian or non-Markovian.

Figure 2

Concurrence [31] [upper-originating (blue) line] of ${\varrho }_{SA}$ is decreasing in every arbitrary time interval [$t$ is in $\hslash \equiv 1$ units]. Hence, deciding about the Markovianity of the dynamics is not possible through this entanglement measure. On the other hand, ${\Delta }_{SA}$ [Eq. (6)] [lower-originating (red) curve] is positive in $t\in [0,6]$, which shows that ${\mathcal{D}}_S\left[{\varrho }_{SE}\left(t\right)\right]$ is nonzero in this interval; hence, the process is non-Markovian.

Figure 3

$S$-discord [lower (red) curve] of example II vs time (in $\hslash \equiv 1$ units) for ${\varrho }_{SE}\left(0\right)=|\phi \rangle \langle \phi |\otimes \left(p\right|0\rangle \langle 0|+(1-p)|1\rangle \langle 1\left|\right)$, where $|\phi \rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$ and $p=0.5$. $S$-discord has been obtained from Eq. (5) in which ${|\psi \left(t\right)\rangle }_{SEA}=(U_{SE}\left(t\right)\otimes \mathbb{1})\left[\right|\phi \rangle \otimes {\left(\sqrt{p}\right|00\rangle }_{EA}+\sqrt{1-p}{|11\rangle }_{EA}\left)\right]$. The upper (blue) curve shows concurrence between the system and ancilla vs time. Since concurrence increases only after $t_{*}=3$, the non-Markovianity property of the dynamics for $t<t_{*}$ is not captured by using the entanglement measure of non-Markovianity.