Multiplexing scheme for simplified entanglement-based large-alphabet quantum key distribution

We propose a practical quantum cryptographic scheme which combines high information capacity, such as provided by high-dimensional quantum entanglement, with the simplicity of a two-dimensional Clauser-Horne-Shimony-Holt (CHSH) Bell test for security verification. By applying a state combining entanglement in a two-dimensional degree of freedom, such as photon polarization, with high-dimensional correlations in another degree of freedom, such as photon orbital angular momentum (OAM) or path, the scheme provides a considerably simplified route towards security verification in quantum key distribution (QKD) aimed at exploiting high-dimensional quantum systems for increased secure key rates. It also benefits from security against collective attacks and is feasible using currently available technologies.

Figure 1

Schematic diagram for a suggested implementation of the proposed simplified large-alphabet entanglement-based quantum key distribution using OAM and polarization. (a) Preparation of the two-photon state $|{\mathrm{\Phi }}_{\varepsilon }\rangle$ [Eq. (3)] or $|{\mathrm{\Phi }}_{\varepsilon }^d\rangle$ [Eq. (8)] using spontaneous parametric down conversion (SPDC) in a $\beta$-barium borate (BBO) nonlinear crystal cut for type-I spontaneous parametric down conversion. The preparation uses an OAM parity sorter [25], a polarizing beam splitter (PBS), and a nonpolarizing 1:1 beam splitter (BS). (b) Measurement setup for Alice (Bob). Measurements 1, 2, and 3, i.e., $A_j$ and $B_k$ ($j,k=1,2,3$) are respectively selected randomly (e.g., using beam splitters) on Alice's and Bob's side. For the Bell test, Alice (Bob) adds (subtracts) $\hslash$ of OAM for vertically polarized photons using PBS1, SLM1 ($\mathrm{OAM}\pm 1$), and PBS2. Then Alice (Bob) sets the half-wave plate (HWP2) orientation angle to implement the randomly chosen measurement ($A_1/B_1$ or $A_2/B_2$). HWP2 and PBS3 are used for polarization analysis in the Bell test, after which Alice (Bob) may then choose to reverse the first operation by using SLM2 ($\mathrm{OAM}\mp 1$) and PBS4. OAM sorting [13] is used to resolve the qubit subspaces and/or establish the key (measurement 3).

Figure 2

Schematic diagram for a deterministic implementation of the proposed QKD scheme (a) Suggested preparation of the source state using an array of single-entangled-photon-pair sources (EPS), spatial light modulators (SLMs), splitters, and OAM combiners. (b) Suggested measurement setup for Alice (Bob). Measurements 1, 2, and 3, i.e., $A_j$ and $B_k$ ($j,k=1,2,3$), are respectively selected randomly (e.g., using beam splitters) on Alice's and Bob's side. For the Bell test, Alice (Bob) sets the half-wave plate (HWP3) orientation angle to implement the randomly chosen measurement. HWP3 and PBS6 are used for polarization analysis in the Bell test. As in Fig. 1, OAM sorting is used to resolve the qubit subspaces and/or establish the key.

Figure 3

Comparison of the minimum secure key rate $r_{\min }$ as a function of the Bell parameter $S$ and the quantum bit-error rate (QBER) $Q$, in a single run, for our scheme with a qubit-based E91-type protocol [17, 18]. We assume that the quantum bit-error rate $Q$ and Bell parameter for each channel is the same, i.e., $Q_n=Q$ and $S_n=S$ respectively for all $n$. Our scheme shows a $d$-fold enhancement in secure key rate.