Spin-orbit-coupled quantum memory of a double quantum dot

The concept of quantum memory plays an incisive role in the quantum information theory. As confirmed by the several recent rigorous mathematical studies, the quantum memory inmate in the bipartite system ${\rho }_{AB}$ can reduce the uncertainty about part $B$, after measurements done on part $A$. In the present work, we extend this concept to systems with a spin-orbit coupling and introduce the notion of spin-orbit quantum memory. We self-consistently explore the Uhlmann fidelity, the pre- and the post-measurement entanglement entropy, and the post-measurement conditional quantum entropy of the system with spin-orbit coupling and show that measurement performed on the spin subsystem decreases the uncertainty of the orbital part. The uncovered effect enhances with the strength of the spin-orbit coupling. We study the concept of macroscopic realism introduced by Leggett and Garg [Phys. Rev. Lett. 54, 857 (1985)] and observe that POVM measurements done on the system under the particular protocol are noninvasive. For the extended system, we perform quantum Monte Carlo calculations and consider the reshuffling of the electron densities due to an external electric field.

Figure 1

Schematic representation of the considered double quantum dot system in the presence of an external electric field and spin-orbit coupling. In a quasi-one-dimensional conductive channel, two quantum dots are created and controlled by two local gates: “gate 1” and “gate 2.” The quantum confinement in $y$ direction is strong enough such that the only lowest subband state $n_y=0$ is occupied. The “gate 0” is used to control tunnel junction between the dots (that mimics the changing of the interdot distance). The applied constant electric field, polarized in $x$ direction, is represented by the blue arrow.

Figure 2

Dependence of the von Neumann entropy on the system's and field parameters: (a) Planes describe the pre- (green) and post-measurement (orange) entropies as a function of the spin-orbit coupling strength $\alpha$ and the applied external electric field $E_0$. The effective interdot distance is $\mathrm{\Delta }=0.8d_0$. (b) The difference between pre- and post-measurement entropies of the orbital subsystem $S\left({\widehat{\rho }}^{\mathrm{or}}\right)-S\left({\widehat{\varrho }}_{AB}\right)$ as a function of the spin-orbit coupling $\alpha$, plotted for different interdot distances.

Figure 3

The pair distribution function $\rho (x_1,x_2)$ at zero magnetic and electric fields for various values of the trapping parameter $\beta$. The Rashba constant $\alpha =0.4$. The parameter $\beta$ defines the inverse localization length of wave function. When the localization length exceeds the distance between minima of the trapping potential $V\left(x\right)$ the electronic wave function is delocalized. (a, b) Delocalized pair distribution function for $\beta =1$ and $\beta =3$. With the increase of $\beta$ the potential barrier between the minima of the potential increases and the electrons become localized in the minima of the potential. (c) Localized pair distribution function for $\beta =10$.

Figure 4

The pair distribution function. The applied electric field steers the electronic density to the edge of the sample in the direction of the field. The pair distribution function of the first two particles $\rho (x_1,x_2)$ is centered at the minima of $V\left(x\right)+eEx$ for $E_0=1$. Various values of the trapping parameter are considered $\beta =1,3,10$. The Rashba constant is equal to $\alpha =0.4$.