Quantum non-Markovianity based on the Fisher-information matrix

With the development of quantum-information theory, there has been a flurry of investigations of quantum non-Markovian dynamics, and several significant measures for such dynamics have been proposed from various perspectives, such as the breakdown of dynamical divisibility, increase in the distinguishability between quantum states, increase in correlations between the system and an arbitrary ancillary, and so on. Motivated by the idea of exploiting the information content of parameters encoded in initial states, we propose a conceptually simple and physically intuitive characterization for non-Markovianity with the help of a quantum-Fisher-information matrix. The basic features are illustrated through several examples, and relations with other approaches are elucidated. A hierarchial aspect of quantum non-Markovianity is revealed.

Figure 1

Graphs of $\mathrm{BLP}\left(\alpha \right)$, $\mathrm{FI}\left(\alpha \right),$ and ${\mathrm{LFS}}_0\left(\alpha \right).$ Note that $\mathrm{BLP}\left(\alpha \right)=\mathrm{FI}\left(\alpha \right)<{\text{LFS}}_0\left(\alpha \right)$ for $\alpha \in (0,1/3),$ while $\mathrm{BLP}\left(\alpha \right)<\mathrm{FI}\left(\alpha \right)<{\text{LFS}}_0\left(\alpha \right)$ for $\alpha \in (1/3,1/2],$ and these three curves coincide when $\alpha =0$ and $\alpha =1/3$. All quantities are dimensionless.

Figure 2

Graph of $h\left(\alpha \right)$. The quantities are dimensionless.