Effects of atmospheric turbulence on the entanglement of spatial two-qubit states

We study the effects of atmospheric turbulence on the entanglement of spatial two-qubit states that are prepared using the signal and idler photons produced by parametric down-conversion. Such states are the basic ingredients of quantum information protocols and can be prepared, for example, by making down-converted photons pass through a pair of double-apertures. We make use of the Kolmogorov model for atmospheric turbulence and quantify the entanglement of the two-qubit state in terms of Wootters’s concurrence. We restrict our analysis to the two-qubit states that can be represented by density matrices having only two nonzero diagonal elements.

Figure 1

A generic scheme of the preparation of spatial two-qubit states. The state of the two photons through the four apertures is represented by the density matrix ${\rho }_{\mathrm{qubit}}$ of Eq. (1). The turbulence-induced wavefront errors are located primarily near the plane of the apertures that define the qubit space.

Figure 2

Plot of the concurrence $C({\rho }_{\mathrm{qubit}})$ (a) versus $\Delta {\rho }^{\prime }/r_0(z)$ for $d/r_0(z)=10$ and (b) versus $d/r_0(z)$ for $\Delta {\rho }^{\prime }=0.1$. Here $d$ is a measure of the separation between the two signal and two idler apertures, whereas $\Delta {\rho }^{\prime }$ can be taken as a measure of the effective physical size of the two-qubit state. Note that the concurrence depends much more strongly on $\Delta {\rho }^{\prime }$ than on $d$.