Spatial Mode Encoding for Quantum Key Distribution: From Hundreds to Thousands of Modes

Here, we present a proof-of-principle high-dimensional quantum key distribution (QKD) protocol utilizing the position and momentum entanglement of photon pairs. The protocol exploits the fact that position and momentum form mutually unbiased bases, linked via a Fourier transform. One photon of the entangled pair is measured by the sender in a randomly chosen basis-either position or momentum-selected passively via a beam splitter. This projective measurement remotely prepares the partner photon in a corresponding spatial mode, which is sent to the receiver, who similarly performs a random measurement in one of the two bases. In this implementation, we achieve a photon information efficiency of 5.07 bits per photon using 90 spatial modes, and a maximum bit rate of 0.9 Kb/s with 361 modes. To assess the scalability of this spatial-mode encoding scheme, we theoretically show that using a brighter entangled photon source along with next-generation single-photon cameras - featuring improved quantum efficiency, timing and spatial resolution - this approach could achieve 9 bits per photon at 2000 spatial modes, and a bit rate of over 700 Mb/s at 4400 modes while accounting for finite-key effects. These results quantify the opportunities and performance bounds of spatially encoded, entanglement-based QKD and provide a benchmark for future high-dimensional quantum communication systems.

💡 Research Summary

This paper presents a proof‑of‑concept high‑dimensional quantum key distribution (QKD) protocol that exploits the continuous‑variable entanglement of photon pairs in position and momentum. By discretizing the continuous correlations into a set of spatial modes, the authors obtain two mutually unbiased bases (MUBs)—the position basis (image plane) and the momentum basis (Fourier plane)—which are linked by a Fourier transform. A 50:50 beam splitter randomly routes each photon to one of the two measurement stations, thereby selecting the measurement basis passively without any external random number generator or active modulators. When Alice measures her photon in a chosen basis, the entanglement projects Bob’s photon into the corresponding spatial mode; Bob then measures his photon using an identical passive basis‑selection setup.

The detection is performed with time‑tagging single‑photon cameras that provide spatially resolved coincidence data in both bases. In the experimental implementation, the authors achieve a photon‑information efficiency of 5.07 bits per detected photon using 90 spatial modes (d = 90) and a maximum sifted key rate of 0.9 kb/s with 361 modes (d = 361). The performance is limited primarily by the camera’s quantum efficiency (~8 %), spatial resolution (256 × 256 pixels), and timing resolution (~8 ns), which together yield an overall system efficiency of about 2 %. Background coincidences and non‑uniform mode probabilities also contribute to the observed quantum dit error rate (QDER).

To assess scalability, the authors model the system with parameters expected for next‑generation superconducting nanowire single‑photon cameras (quantum efficiency > 90 %, picosecond timing, megapixel arrays) and an asymmetric 90:10 beam splitter that biases basis selection toward the encoding basis. Under these assumptions, the overall system efficiency rises to ~60 %, the coincidence window shrinks to 100 ps, and spatial resolution improves by a factor of 1.5. Theoretical calculations using the high‑dimensional BBM92 secret‑key formula R = log₂(d) − 2 Hd(e) show that, with a quantum dit error rate kept below the dimension‑dependent security threshold, the photon‑information efficiency could reach ≈ 9 bits per photon at d ≈ 2000. Correspondingly, a sifted key rate of ~10 Mb/s is predicted for d ≈ 4400.

Further gains are possible by increasing the brightness of the SPDC source. The current experiment uses a 40 mW pump and a 1 mm Type‑I ppKTP crystal. Switching to a Type‑0 phase‑matched configuration, employing higher pump powers, and using a thicker nonlinear crystal could increase the pair‑generation rate by more than 50‑fold. Simulations indicate that a 100‑fold increase in source brightness, combined with the advanced detectors, would enable sifted key rates exceeding 700 Mb/s at d ≈ 4400.

The security analysis also incorporates finite‑key effects, employing a refined key‑rate expression that accounts for the number of bits allocated to raw key generation and parameter estimation, as well as chosen security parameters (ε_sec, ε_cor). Even with finite‑key corrections, the projected performance remains well within practical regimes, provided the error rates stay low.

In summary, the work demonstrates that high‑dimensional spatial‑mode encoding of entangled photons can be realized with fully passive state preparation and measurement, achieving multi‑bit information per photon and offering a clear pathway to gigabit‑per‑second quantum‑secure communication. The paper outlines concrete hardware advancements—superconducting nanowire cameras, brighter SPDC sources, and asymmetric basis selection—that together could transform laboratory‑scale demonstrations into deployable high‑speed QKD systems, thereby advancing the development of future quantum networks.