Entangling protocols due to non-Markovian dynamics

It is widely spread in the literature that non-Markovianity (NM) may be regarded as a resource in quantum mechanics. However, it is still unclear how and when this alleged resource may be exploited. Here, we study the relationship between NM and quantum optimal control under the objective of generating entanglement within $M$ noninteracting subsystems, each one coupled to the same non-Markovian environment. Thus, we design a variety of entangling protocols that are only achievable due to the existence of the environment. We show that NM plays a crucial role in all the entangling protocols considered, revealing that the degree of NM completely determines the success of the entangling operation performed by the control. This is a demonstration of the virtues of NM and the way that it can be exploited in a general entangling setup.

Figure 1

Schematic representation of the entangling setup considered under non-Markovian dynamics. The existence of a backflow of information from the environment to the system makes possible the entanglement between the subsystems by only controlling one of them with an external field $\lambda \left(t\right)$. On the other hand, under a Markovian regime where information flows in one direction from the system to the environment it will not be possible to generate any arbitrary entangling transformation with accuracy.

Figure 2

Driving to a GHZ state ($M=3$). (a) Fidelity achieved in the space of parameters given by the coupling $A$ and the driving time $T$ ($\hslash =1$ and in units of $1/{\omega }_0$). (b) NM of the free evolution.

Figure 3

(a) $M=2$: driving to the Bell state ${|{\mathrm{\Phi }}^{+}\rangle }_S$. (b) $M=3$: driving to the GHZ state. In both panels we plot the fidelity (with control) as a function of the original NM, merging seven different protocols with different couplings and number of environmental spins. While the blue dashed curve corresponds to a dynamics in which $N$ is fixed but the coupling varies ($0\le A\le 0.2$), in the six other cases coupling is fixed but $N$ varies ($M\le N\le 8$). In all protocols, evolution time is fixed in $T=10$. The inset of the upper panel shows the fidelity (without control) as a function of time between the same initial state of the open system ($M=2$) evolved with two “different” dynamics that possess the same degree of NM ($\text{NM}=0.43$). One state is evolved with $N=8$ and $A=0.2/\sqrt{6}$ [i.e., the last red cross from panel (a)] while the other with $N=5$ and $A=0.1466$. Despite having the same original NM, fidelity between both evolved states fluctuates as a function of time.

Figure 4

Driving to an arbitrary Bell state ($M=2$). We plot the optimal concurrence achieved as a function of NM. Fixing as a target each Bell state and fixing the number of total spins (within the interval $M\le N\le 6$), we choose randomly 30 different values for the coupling strength ($0\le A\le 0.2$). Blue markers represent the target state $|{\mathrm{\Phi }}^{+}\rangle$, red $|{\mathrm{\Psi }}^{+}\rangle$, green $|{\mathrm{\Psi }}^{-}\rangle$, and yellow $|{\mathrm{\Phi }}^{-}\rangle$. On the other hand, circles represent a configuration with $N=3$, stars with $N=4$, squares with $N=5$, and triangles with $N=6$. In all protocols, evolution time is fixed in $T=10$.