High-dimensional coherent one-way quantum key distribution

TL;DR

This work introduces a high-dimensional quantum key distribution protocol that achieves arbitrary qudit dimensions using only standard two-dimensional hardware, and provides security proofs against both individual and coherent attacks. The scheme relies on random permutation of time-bin blocks and a Hadamard-based measurement to bound Eve’s information without requiring a full $d$-port interferometer. The authors derive an analytic upper bound for individual attacks and a computable lower bound for coherent attacks via semidefinite programming, then demonstrate a 32-dimensional implementation over 40 km of fiber in a conventional COW setup, achieving more than a twofold increase in the secure key rate in the detector-saturation regime. The results suggest practical scalability across time-bin, spatial, and spectral encodings and point to software-enabled upgrades that can boost existing QKD deployments without hardware changes.

Abstract

High-dimensional quantum key distribution (QKD) offers secure communication, with secure key rates that surpass those achievable by QKD protocols utilizing two-dimensional encoding. However, existing high-dimensional QKD protocols require additional experimental resources, such as multiport interferometers and multiple detectors, thus raising the cost of practical high-dimensional systems and limiting their use. Here, we present and analyze a novel protocol for arbitrary-dimensional QKD, that requires only the hardware of a standard two-dimensional system. We provide security proofs against individual attacks and coherent attacks, setting an upper and lower bound on the secure key rates. Then, we test the new high-dimensional protocol in a standard two-dimensional QKD system over a 40 km fiber link. The new protocol yields a two-fold enhancement of the secure key rate compared to the standard two-dimensional coherent one-way protocol, without introducing any hardware modifications to the system. This work, therefore, holds great potential to enhance the performance of already deployed time-bin QKD systems through a software update alone. Furthermore, its applications extend across different encoding schemes of QKD qudits.

Figures (9)

• Figure 1: Protocol scheme with arbitrary modes. Alice encodes three 4-dimensional qudits, represented here using 12 dual-rail optical modes, and applies a secret random permutation ($\sigma$). Bob is restricted to measuring the photon occupation in each mode and applying a Hadamard ($H$) transformation. Importantly, since Bob does not use a discrete Fourier transform (DFT) measurement, he can use a standard two-dimensional system with a two-port interferometer. Eve attacks the quantum channel by applying a nonlocal unitary transformation $U$. The transformation $U$ operates on a Hilbert space containing the encoded qudits space and an arbitrary-dimension ancilla qudits space, while $\sigma$, $H$, and $DFT$ operate on the optical modes space.
• Figure 2: Example of the protocol scheme implemented with time-bin encoding. In 2-dimensional encoding ($d=8$), Alice transmits $n=1000$ two-dimensional time-bin encoded signals within a time window of 2000 ns, which corresponds to the detector's dead time. At the data line, Bob measures one photon per dead time, yielding 1 bit of mutual information. Additionally, at the monitoring line, Bob measures one photon per dead time, corresponding to a coherence test of one pair of pulses. In 8-dimensional encoding ($d=8$), Alice transmits $n=250$ signals per dead time to Bob, after permuting the time-bins order. Alice may apply the permutation in the software before the generation of the optical pulses. In the data line, Bob measures one photon per dead time yielding 3 bits of mutual information after Alice reveals the permutation. Irrespective of the encoding dimension, Bob employs the same monitoring line to detect potential attacks.
• Figure 3: Bound of the secure key rate for different dimensions. The lower bound (solid lines) and the upper bound (dashed lines) on the secure key rate on a logarithmic scale are presented versus the total system loss in dB. Different colors represent different dimensional encoding sizes. A finite secure key rate lower bound is achieved for dimension $2,3,4$, indicating that the protocol is secure for these dimensions. The optimal lower bound secure key rate is achieved for $d=3$ in the range of up to 8 dB loss. The upper bound is of the same order of magnitude as the lower bound, indicating that it represents an effective attack. Here Bob’s detector dark count rate is $5\cdot10^{-8}$, bit error rate of $0.5\%$, visibility of $99.5\%$, dead time of $10us$, and a pulse duration of $2ns$.
• Figure 4: Experimental setup for comparing arbitrary-dimensional QKD schemes. Alice’s transmitter (left) consists of a continuous wave (CW) laser at $\lambda=1550 nm$ that is modulated using an electro-optic intensity modulator (IM) running at $500 MHz$. The pulses are passed through a variable optical attenuator (VOA) that regulates the mean photon number per pulse. The weak coherent pulses are delivered to Bob's end through a $40 km$ long single-mode fiber (SMF-28). Bob’s receiver (right) consists of an asymmetric beamsplitter, which provides a passive choice of the measurement basis; $90\%$ of the photons travel directly to the data detector, and $10\%$ pass through an unbalanced interferometer and are detected by the monitor detector. We lock the laser's wavelength to the interferometer so that the monitor detector always measures the dark port of the interferometer. The interference visibility is estimated by registering the detection events due to the interfering and non-interfering events. In addition to the $40 km$ long quantum channel that delivers the weak coherent pulses, we use a separate $40 km$ SMF-28 fiber for all classical communication between Alice and Bob and to distribute an optical clock signal between them based on the White Rabbit protocol lipinski2011white. State preparation and sifting are run by two field-programmable gate arrays (FPGA) at Alice's and Bob's ends.
• Figure 5: Secure key rates for different dimensions. Secure bits per second is achieved by multiplying the raw bits per photon by measured photons per second. The secure bits per second is obtained by measuring the bit error rate, visibility, and loss, and applying the secure key rate upper bound found in Eq. \ref{['upper_bound']}. Solid lines represent the result obtained in our experimental system and dashed lines represent our model system. The model assumes the error rate per time bin and the visibility are independent of the dimensionality. The optimum is achieved at $d=8$, where the number of secure bits per second increases by $2.04$ for our system, and by $1.89$ for the model.