Two-qubit non-Markovianity induced by a common environment

We study non-Markovianity as backflow of information in two-qubit systems. We consider a setting where, by changing the distance between the qubits, one can interpolate between independent reservoir and common reservoir scenarios. We demonstrate that non-Markovianity can be induced by the common reservoir and single out the physical origin of this phenomenon. We show that two-qubit non-Markovianity coincides with instances of nondivisibility of the corresponding dynamical map, and we discuss the pair of states maximizing information flowback. We also discuss the issue of additivity for the measure we use and in doing so give an indication of its usefulness as a resource for multipartite quantum systems.

Figure 1

(a) The independent reservoir, $D=200L$. (b) The common reservoir case, $D=4L$. Time-dependent coefficients ${\gamma }_1\left(t\right)$ (blue dashed line) and ${\gamma }_2\left(t\right)$ (blue solid line) are shown in the first column, with ${\gamma }_1\left(t\right)+{\gamma }_2\left(t\right)$ (red dashed line) and ${\gamma }_1\left(t\right)-{\gamma }_2\left(t\right)$ (red solid line) in the second column. The scattering length is chosen to be $0.02a_{\text{Rb}}<a_B^{\text{crit}}$ to ensure the one-qubit channel is Markovian for all times $t$ and $a_{\text{Rb}}$ is the natural scattering length of a rubidium condensate; see the Appendix. Insets: zoom on small values of the decay rates.

Figure 2

Non-Markovianity measure ${\mathcal{N}}_{\text{BLP}}$ (the largest value in each column of the plot) as a function of distance $D/L$ between qubits, for scattering lengths (a) $0.02a_B^{\text{Rb}}<a_B^{\text{crit}}$ (locally Markovian dynamics) and (b) $0.50a_B^{\text{Rb}}>a_B^{\text{crit}}$ (locally non-Markovian dynamics). The graph shows 20 000 randomly drawn pairs of initial states. The pairs are classified into different categories and color coded accordingly as elaborated in the key on the right. The layers of dense points are given in order (starting from the bottom): separable, mixed, pure and mixed, pure states, and maximally entangled states.

Figure 3

Non-Markovianity measure ${\mathcal{N}}_{\text{BLP}}$ (the largest value in each column of the plot) as a function of scattering length $a_B$ between qubits for separation (a) $D=200L$ (independent reservoir) and (b) $D=4L$ (common reservoir). The graph shows 20 000 randomly drawn pairs of initial states. The pairs are classified into different categories and color coded accordingly as elaborated in the key on the right. The layers of dense points are given in order (starting from the bottom): separable, mixed, pure and mixed, pure states, and maximally entangled states.

Figure 4

Maximizing states: non-Markovianity measure ${\mathcal{N}}_{\text{BLP}}=\max \{{\mathcal{N}}_{\Phi },{\mathcal{N}}_{\Psi }\}$ (the highest value at each instance) as a function of distance between qubits, $D$, for scattering lengths (a) $0.02a_B^{\text{Rb}}<a_B^{\text{crit}}$ (locally Markovian dynamics) and (b) $0.50a_B^{\text{Rb}}>a_B^{\text{crit}}$ (locally non-Markovian dynamics). The red solid line represents ${\mathcal{N}}_{\Phi }$ and the blue dashed line represents ${\mathcal{N}}_{\Psi }$.

Figure 5

Additivity of the non-Markovianity measure: ${\mathcal{N}}_{\Phi }$ (red solid line), ${\mathcal{N}}_{\Psi }$ (blue dashed line), and 2${\mathcal{N}}_1$ (black solid line) as a function of scattering length $a_B$, for qubit separations of (a) $D=200L$, (b) $D=20L$, and (c) $D=4L$. Recall that ${\mathcal{N}}_2={\mathcal{N}}_{\text{BLP}}=\max \{{\mathcal{N}}_{\Phi },{\mathcal{N}}_{\Psi }\}$.