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

TL;DR

The results quantify the opportunities and performance bounds of spatially encoded, entanglement-based QKD and provide a benchmark for future high-dimensional quantum communication systems.

Abstract

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.

Figures (7)

• Figure 1: Conceptual setup for position-momentum QKD. Position-momentum entangled photon pairs with orthogonal polarization are created via Type-II SPDC by Alice who keeps the vertically polarized idler photon locally and sends the horizontally polarized signal photon to "Bob". For detection at the two parties, the photons are randomly split to be measured in one of the two MUBs, either in position or momentum, by time-tagging cameras. Finally, the two parties compares their measurement bases via a classical channel to create their secret key. Inset on the top left shows the measured position and momentum correlations in the horizontal $x$ and $u$ direction, correlations in the vertical $y$ and $v$ directions looks near identical. Note that due to possessing only a single camera, in our experiment we subdivided the camera into four regions to act as the four cameras. See Supplementary Material for further information.
• Figure 2: Joint detection matrix $C_{r,k}$ for $d=545$.$C_{r,k}$ exhibits a clear diagonal structure, indicating strong correlations between Alice’s and Bob’s measurement outcomes when they choose the same basis—these events contribute to the shared secret key. In the ideal case with no errors $C_{r,k}$ would be an identity matrix. Insets show the selected pixel layouts for the position $x$ and momentum $k$ beams, with Alice's measured beam on the left and Bob's on the right.
• Figure 3: Effect of increased dimensionality on the sifted key rate and the Qdit error rate (QDER). Plot of the QDER (top), Photon efficiency (middle) and Bit rate (bottom) vs Dimension. The maximum error rate at which QKD could theoretically be implemented for each dimension is shown in the gray area of the top QDER plot and is denoted as the "Security Threshold". The theoretical performance with current experimental parameters is shown as the green curve and compared to the experimentally measured performance, shown in purple. The cyan curve is the expected performance under finite key length while using next generation detectors as well as having a 90:10 beam splitter for MUB selection instead of 50:50.
• Figure 4: Effect of increased source brightness on the sifted key rate The expected sifted key rate in the finite-key regime with increased source brightness using next generation detectors.
• Figure 5: (a) Experimental setup for position-momentum QKD. LP-filter: long-pass filter, BP-filter: band-pass filter, PBS: polarizing beam-splitter, BS: beam-splitter (b) Image captured on camera of the position and momentum planes of SPDC.