Secure information capacity of photons entangled in many dimensions

We quantify precisely the maximum secure information capacity of photons entangled in high dimensions for entanglement in the orbital angular momentum and angular degrees of freedom. Our analysis takes careful account of the influence of experimental imperfections, such as nonunity detection efficiency, on the degree of Einstein-Podolsky-Rosen (EPR) entanglement and hence on the secure information capacity of the photon pairs. We find that there is is an optimal dimension that maximizes the secure information capacity whose value can be predicted analytically from the knowledge of only a few experimental parameters.

Figure 1

ND, neutral density filter; BBO, $\beta$ barium borate crystal; SLM, spatial light modulator; IF, interference filter; $L_1$, 150 mm; $L_2$, 500 mm; $L_3$, 1000 mm; and $L_4$, 1.45 mm. The combination of the lenses $L_1$ and $L_2$ ($L_3$ and $L_4$), act as a $4f$ imaging system; thus the distance from the crystal to each SLM is 1300 mm (the distance from the SLMs to the fibers is $\sim$2003 mm).

Figure 2

Coincidence count rates for the OAM basis (a) and the angle basis (b) for the case of $D=11$. For this data set, there is no neutral density filter placed between the pump and the BBO crystal. The integration time for each measurement point is 1 s.

Figure 3

Inferred entropy $H_{\mathrm{Inf}}\left({\ell }_B\right)+H_{\mathrm{Inf}}\left({\phi }_B\right)$ as a function of dimension $D$ and pair generation rate. The green squares represent the conditions where inequality (5) is violated and hence the secure information is greater than zero; the red circles represent the conditions where it is not violated and hence there can be no secure information. The gray dashed lines are the theoretical predictions based on our model.

Figure 4

Contour plots of the maximum secure information capacity $\Delta I$ and information rate $R_{\Delta I}$ ($\eta =5\%$, $P_X=10\%$, and $\Delta t=25$ ns). The red dotted lines indicate the threshold to achieve a positive $\Delta I$ and $R_{\Delta I}$, the green dashed line indicates the analytical solution for the optimal dimension $D_{\mathrm{opt}}$ given singles rate $S$, and the blue solid lines indicate the numerically found maxima.